Nov 25, 2010 - 5.1.1 Down-Conversion and IF Signal Processing . . . . . . . . . . 128 .... 72. 3.13 PSD and autocorrelation function for low-pass process. . . . . . . . . 73 xi ...... ubx-g6010-gps-single-chip.html, [Accessed: 01 July 2010]. [114] A. J. ...
Weak Signal Digital GNSS Tracking Algorithms James T. Curran November 25, 2010
A Thesis Submitted to the National University of Ireland in Fulfilment of the Requirements for the Degree of Ph.D.
Supervisors: Dr. Colin C. Murphy & Prof. G´erard Lachapelle Head of Department: Dr. Michael Creed Department of Electrical and Electronic Engineering, National University of Ireland, Cork.
ii
Abstract The design of weak-signal GNSS carrier tracking algorithms is addressed in this thesis. Particular emphasis is placed on the consumer grade receiver and the civilian use of GNSS. The challenges posed by both the limited resources afforded by this receiver, and by the harsh operating environments encountered in the civilian use of GNSS, are considered. A novel analysis of the impact of the receiver front-end filter and quantiser is presented, resulting in new expressions for the related signal-to-noise ratio loss in the DMF. It is also shown that a joint analysis of the effects of the front-end filter and the quantiser is necessary to accurately predict this loss. This theory is applied to the problem of receiver front-end design and novel results are presented pertaining to the choice of front-end filter bandwidth for various receiver configurations. New, exact, closed form expressions and approximate expressions are developed for the design of first- and second-order tracking loops, respectively, which facilitate the accurate design of tracking loops implemented in a digital receiver. Novel closed from expressions for the tracking error variance of these loops are also developed. The characteristics of four popular carrier frequency estimators are examined and new results are presented, describing their operation in a weak signal environment. These results are employed in an analysis and optimisation of FLL design across a range of received signal-to-noise ratios. Novel design rules are presented which can achieve in excess of a 10 dB improvement over a n¨aive FLL design. The properties of four popular carrier phase estimators are examined under weak signal conditions and new insight into the implications of carrier phase estimator choice on PLL performance is provided. Both the linear, steady-state and the nonlinear, transient operation of the PLL is considered in this analysis. Local oscillator induced phase tracking error variance is examined and optimal, Weiner filter based, PLL filters are developed, conditioned on the characteristics of the oscillator and thermal noise, providing MMSE tracking performance for the GNSS PLL. The theoretical results presented in this thesis are validated through both simulated, Monte-Carlo, experiments and through post-processing of actual GPS L1 C/A signals.
iii
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Acknowledgments I would like to thank my supervisor, Dr. Colin C. Murphy, for his commitment to this undertaking. In addition to the original inspiration for the project, his continued encouragement, painstaking attention to detail and his unique perspective on this work have been invaluable over the last four years. For facilitating my year of study with the PLAN Group, I would like to thank my co-supervisor, Prof. G´erard Lachapelle. The exposure he afforded me to world-class resources and to talented, enthusiastic researchers, greatly accelerated my research. Particularly, Cillian and Daniele, for your advice, collaboration and encouragement, thank you. Both the computational resources and the technical assistance provided to me by the Boole Centre for Research in Informatics were a treasured asset. I would particularly like to acknowledge the contributions and assistance of Brain Clayton. This project would not have been possible were it not for the financial support of the IRCSET Embark postgraduate scholarship scheme, I am grateful for the opportunity that they have provided. To the staff of the Department of Electrical and Electronic Engineering, thank you. Your support and company have made the last four years a pleasure. To my friends and fellow students in the university, particularly Dave, Eoin, Neil, Tommy, Niamh, Orla, Padraig, Rob, Brian and Steve, your camaraderie is greatly appreciated. To my friends in Calgary, thank you for welcoming me and making me feel immediately at home. Your friendship and kindness has left a lasting impression on me. Finally, I would like to thank my family. To my parents, brother and sister, thank you for your encouragement, patience, support and for the solace of home.
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vi
Contents 1 Introduction
1
1.1
Objectives and Motivaton . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.3
Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2 Introduction to Signals and Receivers
7
2.1
Overview of the GNSS signal structure . . . . . . . . . . . . . . . . .
8
2.2
Propagation Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.2.1
Received Signal Power . . . . . . . . . . . . . . . . . . . . . .
15
2.2.2
Multipath . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.2.3
Thermal Noise . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.2.4
Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.2.5
Doppler and Doppler Drift . . . . . . . . . . . . . . . . . . .
19
2.2.6
Atmospheric Effects . . . . . . . . . . . . . . . . . . . . . . .
20
2.3
The Matched Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.4
Receiver Non-Idealities and Associated Losses . . . . . . . . . . . . .
23
2.4.1
Filtering Losses . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.4.2
Quantisation Losses . . . . . . . . . . . . . . . . . . . . . . .
26
2.4.3
The Local Oscillator . . . . . . . . . . . . . . . . . . . . . . .
29
Baseband Receiver Processing . . . . . . . . . . . . . . . . . . . . . .
36
2.5.1
The Frequency Lock Loop . . . . . . . . . . . . . . . . . . . .
39
2.5.2
The Phase Lock Loop . . . . . . . . . . . . . . . . . . . . . .
42
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
2.5
2.6
3 Quantisation and Filtering Losses
49
3.1
Signal and System Model . . . . . . . . . . . . . . . . . . . . . . . .
50
3.2
The Ideal DMF and The Loss Coefficient . . . . . . . . . . . . . . .
54
3.3
Front-End Filtering Effects . . . . . . . . . . . . . . . . . . . . . . .
59
3.3.1
Propagation of the Mean . . . . . . . . . . . . . . . . . . . .
59
3.3.2
Propagation of the Variance . . . . . . . . . . . . . . . . . . .
63
3.3.3
Filtering Losses . . . . . . . . . . . . . . . . . . . . . . . . . .
66
Quantisation Effects . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
3.4
vii
Contents 3.4.1
Propagation of the Mean . . . . . . . . . . . . . . . . . . . .
68
3.4.2
Gaussian Sequences and Non-linear Functions . . . . . . . . .
71
3.4.3
Propagation of the Variance . . . . . . . . . . . . . . . . . . .
75
3.4.4
Joint Quantisation and Filtering Loss . . . . . . . . . . . . .
78
3.5
Front-end Filter Optimisation . . . . . . . . . . . . . . . . . . . . . .
80
3.6
Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
3.6.1
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . .
87
3.6.2
Real Signal Validation . . . . . . . . . . . . . . . . . . . . . .
88
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
3.7
4 Analysis and Design of Frequency Lock Loops 4.1
93
Problem Statement and System Architecture . . . . . . . . . . . . .
94
4.1.1
Linear Model of the FLL . . . . . . . . . . . . . . . . . . . .
95
4.1.2
Noise Bandwidth and Tracking Error Variance . . . . . . . .
97
4.2
Design Equations for The Loop Filter . . . . . . . . . . . . . . . . .
99
4.3
Discriminator Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.3.1
The Four-Quadrant Arctangent Discriminator . . . . . . . . . 103
4.3.2
The Arctangent Discriminator . . . . . . . . . . . . . . . . . 109
4.3.3
The Cross Product Discriminator . . . . . . . . . . . . . . . . 110
4.3.4
The Decision Directed Cross-Product Discriminator . . . . . 113
4.3.5
Approximations For The Arctangent Discriminators . . . . . 117
4.4
Impact of KD on Transient Response . . . . . . . . . . . . . . . . . . 118
4.5
Steady-State Performance of the FLL . . . . . . . . . . . . . . . . . 119 4.5.1
Tracking Error Variance . . . . . . . . . . . . . . . . . . . . . 119
4.5.2
Comparisons of Steady-State Performance . . . . . . . . . . . 121
4.6
A note on SNRc
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.7
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5 The Phase Lock Loop I 5.1
5.2
5.3
127
Receiver Model and PLL Architecture . . . . . . . . . . . . . . . . . 128 5.1.1
Down-Conversion and IF Signal Processing . . . . . . . . . . 128
5.1.2
The Phase-Lock Loop . . . . . . . . . . . . . . . . . . . . . . 129
Carrier Phase Estimation . . . . . . . . . . . . . . . . . . . . . . . . 132 5.2.1
The four-quadrant arctangent discriminator . . . . . . . . . . 132
5.2.2
The arctangent discriminator . . . . . . . . . . . . . . . . . . 136
5.2.3
The decision-directed (Costas) discriminator . . . . . . . . . 138
5.2.4
The quadrature discriminator . . . . . . . . . . . . . . . . . . 139
5.2.5
The discriminator Linear Region . . . . . . . . . . . . . . . . 140
Closed Loop Performance: Linear Operation . . . . . . . . . . . . . . 142 5.3.1
Impact of KD on Transient Response
5.3.2
Thermal Noise Induced Tracking Error and the GNR . . . . . 148 viii
. . . . . . . . . . . . . 142
Contents 5.3.3 5.4
5.5
The Optimum Discriminator for Linear Operation . . . . . . 151
Closed Loop Performance: Non-Linear Operation . . . . . . . . . . . 153 5.4.1
The PLL Acquisition Stage . . . . . . . . . . . . . . . . . . . 154
5.4.2
Time To Settle (TTS) . . . . . . . . . . . . . . . . . . . . . . 157
5.4.3
The Experiment . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.4.4
The Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.4.5
Preferred Discriminator for the Acquisition Stage . . . . . . . 162
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6 The Phase Lock Loop II
171
6.1
The Local Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.2
Design of Optimal PLL Filters . . . . . . . . . . . . . . . . . . . . . 177
6.3
6.2.1
Wiener Filter Design . . . . . . . . . . . . . . . . . . . . . . . 178
6.2.2
White Frequency Noise . . . . . . . . . . . . . . . . . . . . . 182
6.2.3
Limitations of a First Order Loop . . . . . . . . . . . . . . . 184
6.2.4
White and Random Walk Frequency Noise . . . . . . . . . . 184
6.2.5
Numerical Example . . . . . . . . . . . . . . . . . . . . . . . 187
6.2.6
The case for a 3rd order PLL . . . . . . . . . . . . . . . . . . 187
Test and Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 6.3.1
Real Signal Validation of Theoretical Model . . . . . . . . . . 189
6.3.2
Comparison with Traditional Designs . . . . . . . . . . . . . 191
6.4
Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.5
Towards an Adaptive PLL . . . . . . . . . . . . . . . . . . . . . . . . 196
6.6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
7 Conclusions and Recommendations for Future Work
201
7.1
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
7.2
Recommendations for Future Work . . . . . . . . . . . . . . . . . . . 204
7.3
Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
A Approximations For The FLL Discriminators
209
B Approximations For The PLL Discriminators
213
C Influence of KD on Tracking Error Variance
215
ix
x
List of Figures 2.1
Composition of a BPSK DSSS waveform. . . . . . . . . . . . . . . .
9
2.2
Spreading code chip materialisation. . . . . . . . . . . . . . . . . . .
11
2.3
GNSS chip autocorrelation
. . . . . . . . . . . . . . . . . . . . . . .
13
2.4
Spreading code autocorrelation. . . . . . . . . . . . . . . . . . . . . .
14
2.5
Multipath Propagation . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.6
Doppler Variation over Time . . . . . . . . . . . . . . . . . . . . . .
20
2.7
Block diagram of a correlator. . . . . . . . . . . . . . . . . . . . . . .
23
2.8
PSD of GNSS signals. . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.9
Front-End Filter Loss . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.10 Quantisation of Analogue Signal . . . . . . . . . . . . . . . . . . . .
27
2.11 Example of Oscillator Phase Noise . . . . . . . . . . . . . . . . . . .
31
2.12 Example of Allan and Hadamard Variance . . . . . . . . . . . . . . .
33
2.13 Oscillator Phase Noise PSD . . . . . . . . . . . . . . . . . . . . . . .
34
2.14 Example Acquisition Search Space. . . . . . . . . . . . . . . . . . . .
38
2.15 Generic FLL Block Diagram . . . . . . . . . . . . . . . . . . . . . . .
39
2.16 FLL Noise Performance . . . . . . . . . . . . . . . . . . . . . . . . .
41
2.17 Generic PLL Block Diagram . . . . . . . . . . . . . . . . . . . . . . .
43
2.18 PLL bandwidth choice . . . . . . . . . . . . . . . . . . . . . . . . . .
46
3.1
Block Diagram of Front-End. . . . . . . . . . . . . . . . . . . . . . .
50
3.2
Block Diagram of Equivalent Front-End Model. . . . . . . . . . . . .
51
3.3
Block Diagram of Correlator. . . . . . . . . . . . . . . . . . . . . . .
52
3.4
In-phase and quadrature correlator mean values and envelope. . . . .
61
3.5
The angle of the correlator mean value. . . . . . . . . . . . . . . . .
62
3.6 |µy | versus δτ Tc1 for a selection of front-end filter bandwidths. . . . 3.7
max |µy | versus Bf . . . . . . . . . . . . . . . . . . . . . . . . . . . .
δτ,δωc
63 64
PSD of the noise component of y rms. . . . . . . . . . . . . . . . . .
66
Normalised correlator variance versus Bf . . . . . . . . . . . . . . . .
67
3.10 Loss coefficient (L) versus Bf . . . . . . . . . . . . . . . . . . . . . . .
68
3.11 Transfer characteristic of the B-bit symmetric quantiser. . . . . . . .
69
3.12 KQ versus Ag σf for the one-, two- and three bit quantiser. . . . . . .
72
3.13 PSD and autocorrelation function for low-pass process. . . . . . . . .
73
3.8 3.9
xi
List of Figures 3.14 Autocorrelation function of the quantised sequence. . . . . . . . . . .
76
3.15 Correlator variance versus Ag σf . . . . . . . . . . . . . . . . . . . . .
78
3.16 Quantisation and filtering loss versus Ag σf . . . . . . . . . . . . . . .
79
3.17 Frequency response of ideal IF filter, Hf
81
ej2πf
. . . . . . . . . . . .
3.18 Region of usable tBf , Fc u pairs for the front-end filter. . . . . . . . .
81
3.19 Loss Surface for One-Bit Quantisation GPS C/A (10 MHz). . . . . .
82
3.20 Loss Surface for Two-Bit Quantisation GPS C/A (10 MHz). . . . . .
82
3.21 Loss Surface for Three-Bit Quantisation GPS C/A (10 MHz). . . . .
83
3.22 Loss Surface for One Bit Quantisation GPS C/A (25 MHz) . . . . .
84
3.23 Loss Surface for Two Bit Quantisation GPS C/A (25 MHz) . . . . .
85
3.24 Loss Surface for Three Bit Quantisation GPS C/A (25 MHz) . . . .
85
3.25 One-Bit Quantisation Loss GIOVE-A E1 B/C (25 MHz). . . . . . .
86
3.26 One-Bit Quantisation Loss GIOVE-B E1 B/C (25 MHz). . . . . . .
87
3.27 One-bit Loss Surface: Simulation vs Theory (Fs = 25 MHz). . . . .
88
3.28 C {N0 Vs Front-End Bandwidth using a real GPS signal. . . . . . . .
89
4.1
Block Diagram of a typical digital frequency lock loop. . . . . . . . .
95
4.2
Linearised FLL Model . . . . . . . . . . . . . . . . . . . . . . . . . .
96
4.3
Four-quadrant arctangent discriminator response. . . . . . . . . . . . 105
4.4
Gain curves for the arctangent discriminators. . . . . . . . . . . . . . 106
4.5
Variance curves for the arctangent discriminators.
4.6
The
4.7
Arctangent discriminator response. . . . . . . . . . . . . . . . . . . . 109
4.8
Cross product discriminator response. . . . . . . . . . . . . . . . . . 112
4.9
The ratio
{ R1n
R0n
. . . . . . . . . . 107
curves for the arctangent discriminators. . . . . . . . . 108
R1n{R0n for the cross-product discriminator. .
. . . . . . . 113
4.10 Decision directed cross product discriminator response. . . . . . . . . 114 4.11 Gain curves for the decision directed cross product discriminator. . . 116 4.12 R1n for the decision directed cross product discriminator . . . . . . . 117 4.13 FLL response to frequency step.
. . . . . . . . . . . . . . . . . . . . 120
4.14 The tracking error variance for each of the four discriminators. . . . 121 4.15 The ratio of tracking error variance for different discriminators. . . . 123 4.16 Tracking performance threshold for various discriminators. . . . . . . 124 5.1
Block Diagram of a Standard PLL . . . . . . . . . . . . . . . . . . . 129
5.2
Linearised PLL Model . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.3
A generalized PI controller
5.4
Response of the four-quadrant arctangent discriminator . . . . . . . 133
5.5
Discriminator gain curves for various discriminators. . . . . . . . . . 134
5.6
Variance curves for various phase discriminators. . . . . . . . . . . . 135
5.7
Response of the arctangent discriminator. . . . . . . . . . . . . . . . 136
5.8
Response of the decision-directed discriminator. . . . . . . . . . . . . 138
. . . . . . . . . . . . . . . . . . . . . . . 131
xii
List of Figures 5.9
Linear region for the four-quadrant arctangent discriminator. . . . . 141
5.10 LR versus SNRc for various discriminators. . . . . . . . . . . . . . . 143 5.11 The response of the PLL to phase and frequency steps. . . . . . . . . 146 5.12 GNR versus SNRc for the various discriminators. . . . . . . . . . . . 150 2 versus SNR for various discriminators. . . . . . . . . . . . . . . 151 5.13 σδθ c
5.14 The ratio of the GNR for various discriminators. . . . . . . . . . . . 152 5.15 PLL discriminator choice for linear operation. . . . . . . . . . . . . . 153 5.16 PLL acquisition example . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.17 Distribution of TTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 5.18 Estimated TTS distribution . . . . . . . . . . . . . . . . . . . . . . . 163 5.19 TTS versus PLL bandwidth . . . . . . . . . . . . . . . . . . . . . . . 164 5.20 Acquisition Discriminator Choice . . . . . . . . . . . . . . . . . . . . 168 6.1
Allan Deviation plots for four oscillators. . . . . . . . . . . . . . . . . 174
6.2
Example of oscillator-induced apparent Doppler for a TCXO. . . . . 176
6.3
Linearised PLL Model used for loop optimisation. . . . . . . . . . . . 178
6.4
Wiener Filter Block Diagram. . . . . . . . . . . . . . . . . . . . . . . 178
6.5
Performance of Wiener filter for real signals. . . . . . . . . . . . . . . 191
6.6
Comparison of Wiener filter and traditional designs. . . . . . . . . . 192
6.7
Wiener filter design point sensitivity analysis. . . . . . . . . . . . . . 195
6.8
A PLL with signal-to-noise ratio based Gain Scheduling . . . . . . . 197
6.9
C {N0 and Bω variation for the adaptive PLL. . . . . . . . . . . . . . 198
6.10 Adaptive and fixed Bω PLL prformance comparison. . . . . . . . . . 199 7.1
Relationship between thesis contributions, signal propagation and receiver algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
xiii
xiv
List of Tables 2.1
MRSP for consumer receiver
. . . . . . . . . . . . . . . . . . . . . .
15
2.2
Propagation Losses of Materials . . . . . . . . . . . . . . . . . . . . .
17
3.1
Signal and Receiver Parameters . . . . . . . . . . . . . . . . . . . . .
60
3.2
Optimum Filter Parameters (Fs = 10 MHz). . . . . . . . . . . . . .
83
3.3
Optimum Filter Parameters (Fs = 25 MHz). . . . . . . . . . . . . .
86
3.4
Optimum Filter Parameters for GIOVE E1-B/C Signals . . . . . . .
86
5.1
PLL Design Parameters for Transient Response Experiment . . . . . 145
5.2
Receiver configuration for TTS simulations. . . . . . . . . . . . . . . 161
6.1
Typical Oscillator h-parameters . . . . . . . . . . . . . . . . . . . . . 173
6.2
Numerical Example of Filter Design . . . . . . . . . . . . . . . . . . 187
6.3
Real signal measured and predicted performance. . . . . . . . . . . . 190
xv
xvi
Nomenclature Acronyms A-GPS ADC AGC AWGN BOC BPSK C/A CDMA CW DDSS DLL DMF FFM FLL FPM GIOVE GNSS GPS ICD IF LAN LNA LOS MLS MMSE MRSP NCO NLOS NRZ OCXO PCM PLL
Assisted GPS Analogue-to-digital converter Automatic gain control Additive white Gaussian noise Binary offset carrier Binary phase shift keyed Coarse / acquisition Code division multiple access Continuous wave Direct-sequence spread-spectrum Delay lock loop Digital matched filter Flicker frequency modulation Frequency lock loop Flicker phase modulation Galileo in-orbit validation element Global navigation satellite system Global positioning system Interface control document Intermediate frequency Local area network Low noise amplifier Line of sight Minimum least squares Minimum mean square error Minimum received signal power Numerically controlled oscillator Non-line-of-sight Non-return-to-zero Oven controlled crystal oscillator Pulse code modulation Phase lock loop xvii
Nomemclature
ppb PSD PVT RF RHCP RWFM TCXO TOA WFM
Parts per billion Two-sided power spectral density Position, velocity and time Radio frequency Right hand circular polarised Random walk frequency modulation Temperature compensated crystal oscillator Time of arrival White frequency modulation
xviii
Nomemclature Symbols Received signal power at the antenna One-sided thermal noise floor in Watts per Hertz (W/Hz) Ratio of total carrier power to the one-sided thermal noise floor in units of Hz (or dB Hz) Coherent correlator signal-to-noise ratio Non-coherent correlator signal-to-noise ratio Two-sided power spectral density of the phase estimate noise Autocorrelation of the frequency estimate noise Variance of the frequency estimate noise, the zeroth sample of Rn rms The first sample of Rn rms
P N0 C {N0 SNRc SNRnc Nθ Rn rms R0n R1n
Signal carrier phase in units of rad Signal carrier frequency in units of rad{s Signal code phase in units of chips Receiver estimate of signal carrier phase in units of rad Receiver estimate of signal carrier frequency in units of rad{s Receiver estimate of signal code phase in units of chips Carrier phase tracking error in units of rad Carrier frequency tracking error in units of rad{s Carrier frequency drift tracking error in units of rad{s2 Code phase tracking error in units of chips Doppler frequency Phase estimate noise Frequency estimate noise Carrier phase tracking error variance in units of rad2 Carrier frequency tracking error variance in units of rad2 s2 Time to settle. Defined as the time taken for a PLL to settle to within a certain range of its final settled phase. Duration of time within which there is an x% probability that a given PLL will have settled.
θ ω τ θˆ ω ˆ τˆ δθ δω δ ω9 δτ ωd nθ nω 2 σδθ 2 σδω TTS TTSx%
Hf ejω hf ejω Bf QBg rxs Fs Ts Tc A
Fourier transform of the equivalent front-end filter Impulse response of the equivalent front-end filter One-sided bandwidth (positive frequencies only) of the frontend filter in Hertz B-bit symmetric quantisation operator with AGC gain Ag Sample rate applied to the IF digitised signal Sample period, the reciprocal of Fs Spreading code chip period xix
Nomemclature
Tcode TI TL
Spreading code period Coherent integration (correlation) period in seconds Tracking loop update period in seconds (typically an integer multiple of TI )
KD D pz q F pz q N COpz q
Discriminator gain Z-domain transfer function of the frequency discriminator Z-domain transfer function of the loop filter Z-domain transfer function of the numerically controlled oscillator Transfer function between tracking error and noise process. Transfer function between tracking error and signal phase. Transfer function between receiver phase estimate and and signal phase. Two-sided equivalent rectangular bandwidth of an FLL in Hertz Transfer function between tracking error and signal frequency. Two-sided equivalent rectangular bandwidth of an PLL in Hertz
Hn pz q Hθ pz q Hθˆ Bω Hω pz q Bθ Ψ ptq ψ pt q T pt q ∆1 T pt; τ q ∆2 T pt; τ q ∆3 T pt; τ q 2 pτ q σA 2 pτ q σH
Total phase process of an oscillator Stochastic phase process of an oscillator Time process of an oscillator The first increment of T ptq The second increment of T ptq The third increment of T ptq The Allan variance The Hadamard variance
xx
Nomemclature Mathematical Functions and Operators < tx u = tx u x |x| =x rxs sgn pxq ~x A tB
P C : Du
pa bq rms
The real part of x The imaginary part of x Complex conjugation operator Magnitude operator The argument (or angle) of x measured anti-clockwise from the positive real axis The next largest integer, not greater than x The sign of x, given by sgn pxq 2upxq 1 The three dimensional vector x A equals values of B in the set C such that condition D is satisfied The convolution of a and b given by: pa bq rms
8 ¸
def
8
a rns b rm ns.
n
F tf ptqu F 1 tF pω qu Z tf rnsu Z 1 tF pz qu PT rX pz qs
The Fourier transform of f ptq The inverse Fourier transform of F pω q The Z-transform of f rns The inverse Z-transform of F pz q The positive-time part of X pz q, equal to the factors of X pz q which have stable causal inverse z-transform
U pa, bq N µ, σ 2
E rxs Var rxs
Continuous uniform distribution with support x P ra, bs Normal (Gaussian) distribution with mean µ and variance σ2 The probability density function of a continuous random variable Expectation operator Variance operator
δk erf pxq erfc pxq upxq j
Kronecker delta function Gauss error function Complimentary error function: erfc pxq 1 erf pxq Unit (or Heaviside) step function Imaginary unit, defined by j 2 1
p p φq
xxi
xxii
Chapter 1
Introduction The use of global navigation satellite systems (GNSS) is rapidly pervading our everyday lives. Once considered to be specialist tools or luxury items, devices capable of enlisting these systems in the provision of time and location information to civilian users can, today, be found in abundance. Although these devices appear ubiquitous, the availability of time and location information, does not. Confronted with the challenges posed by indoor environments, in which we spend the vast majority of our time, the operation of these devices is hampered, at best, and is often completely thwarted. In light of the proposed role of global navigation satellite systems in modernised emergency services, this deficiency is, perhaps, alarming. A solution to this problem, therefore, may no longer be merely desirable; it may, in fact, be necessary.
1.1
Objectives and Motivaton
Following the staggering growth in the availability and consumer uptake of GNSSenabled mobile handsets, due, in part, to technical innovations triggered by the US E911 mandates and, subsequently, the European E112 mandates; the design of GNSS receivers and algorithms has seen a paradigm shift. Upon its conception, designers of early global positioning system (GPS) receivers have enjoyed the benefits of high performance components and dedicated application-specific modules. The recent demand for embedded GNSS capability, however, has imposed severe constraints on receiver design, requiring minimal power consumption, cost and size. This trend has led to the sharing of handset resources (such as the front-end and the signal processing components) between various handset functions, and has also contributed to the adoption of software defined receivers. Furthermore, owing to the typical operating environment of a mobile handset (urban canyons and indoors) the received signal can be heavily attenuated and severely distorted. The culmination of these receiver and environmental effects poses a significant challenge to the receiver algorithms and, typically, results in scenarios in which 1
Chapter 1: Introduction traditional algorithms fail to provide satisfactory location information. Designers of embedded GNSS receivers must, therefore, fully utilise the available signal power and receiver resources, and, so, require a thorough understanding of the receiver non-idealities, operation, capability and error sources. Although the challenge for the designer can be posed in one simple question: “How should the limited resources afforded by a consumer grade receiver be focused upon the task of tracking weak GNSS signals?” its resolution is not so palpable. To grasp the problem, its constituent parts must be identified, each assessed and tackled in succession and, finally, the solutions consolidated to yield a composite design. Eminent amongst these intermediate problems are the questions: i. How does a consumer grade receiver affect the received signal? ii. How do tracking algorithms behave under weak signal conditions? iii. How does a consumer grade receiver affect the tracking algorithms? In response to these challenges, this thesis attempts to analyse, in detail, the receiverinduced losses incurred during the reception of GNSS signals. It examines the behaviour of GNSS carrier synchronisation algorithms when applied to highly attenuated signals, in an effort to quantify these algorithm-based losses. Subsequently, it considers the influence of consumer grade receiver components on the performance of these tracking algorithms. Finally, equipped with a thorough analysis of the receiver operation, it strives to develop mitigating design approaches to alleviate some of these losses, focusing, primarily, on the carrier synchronisation aspect of GNSS signal reception.
1.2
Thesis Outline
The problem of tracking weak GNSS signals is, by no means, a new one. Naturally, efforts to achieve satisfactory receiver functionality in ever more challenging environments have been ongoing since the launch of the first GNSS satellites. To place the work presented here in context, and to offer a benchmark against which innovations presented here can be assessed, Chapter 2 presents a review of current carrier tracking algorithms. The GNSS signal is presented and the nature of its propagation to the receiver is discussed. Characteristic traits of a typical consumer grade receiver, the target platform for tracking algorithms considered in this thesis, are identified and described. Finally, some global simplifying assumptions are detailed to delimit the scope of the results presented here. The objective of this thesis is to examine the process of carrier tracking of weak GNSS signals, under the constraints imposed by consumer grade receivers, with a 2
Section 1.2: Thesis Outline view to enhancing tracking performance through astute receiver design. This thesis strives to achieve this by examining the operations applied to the received signal as it propagates from the receiver’s antenna towards its eventual destination as a carrier parameter estimate. The first facet of the process, which renders the received signal to a form that can be manipulated by the carrier tracking loops, is examined in Chapter 3. Specifically, Chapter 3 considers the filtering, sampling and quantisation of the received signal and its subsequent propagation through the receiver’s digital matched filter. Via a thorough appraisal of this process the resultant degradation of signal quality is quantified. Moreover, it is illustrated that this loss can be reduced through careful front-end design. The digital matched filter provides the receiver with the fundamental measurements from which estimates of the received signal parameters are derived. Of course, ultimately, the more signal parameters that can be estimated, the greater the performance of the tracking algorithm will be. However, initially, the receiver will begin with just two parameters: the carrier frequency and the code phase. Carrier tracking algorithms are the primary focus of this thesis and, so, the process of carrier frequency tracking is examined in Chapter 4. This process is, typically, implemented as a frequency lock loop (FLL), which is a recursive algorithm. It consists of three primary operations: the generation of the raw signal measurements using the DMF and the previous carrier parameter estimates; estimation of the current carrier frequency error; and the filtering of this raw estimate. As the first of these functions has been considered in Chapter 3, carrier frequency error estimation is examined in the first part of Chapter 4. Armed with this analysis, Chapter 4 continues with the development of a comparative study of the closed loop tracking performance achieved by these carrier frequency discriminators. An appropriately designed frequency lock loop, under reasonable operating conditions, will, generally, yield a frequency estimate that converges toward the true carrier frequency of the received signal. With this diminished frequency uncertainty, a receiver can attempt to estimate a further carrier parameter: its phase. Typically, a phase lock loop (PLL) is employed for this task. Homologous to the frequency lock loop, the phase lock loop is a recursive estimator of carrier phase, differing, primarily, in the parameter being tracked. Maintaining a similar structure to Chapter 4, Chapter 5 begins by analysing and modelling some carrier phase discriminators, identifying their key characteristics, and follows by utilising this model in a comparison of the linear closed loop tracking capabilities of these discriminators. This linear model of the phase lock loop, alone, offers limited insight. Indeed, the transition from frequency tracking to phase tracking exercises the non-linear region of the phase discriminators and, thus, effects a non-linear system. The second half of Chapter 5 describes a Monte-Carlo based simulation campaign, designed to 3
Chapter 1: Introduction explore the operation of the phase lock loop throughout this transient phase. Results from this study are presented and form a set of guidelines, for choosing a phase discriminator, which can enhance tracking performance. Having considered the estimation of phase error, attention is turned to its use in the phase lock loop. Generally, this raw estimate is processed to extract further carrier parameter estimates. By observing the carrier phase as it evolves over time, the carrier frequency and frequency drift can be estimated. These estimates of phase are, however, corrupted by thermal noise and so an appropriate filter must be employed which can elicit phase information. Perhaps, then, given some a priori familiarity with the carrier phase evolution, a competent trade-off between phase suppression and phase extraction can be made. Chapter 6 approaches this question from an optimal filtering perspective. Specifically, assuming that instabilities in the receiver local oscillator are the dominant impetus in the evolution of the carrier phase, a Wiener filter is developed to estimate this phase in the presence of thermal noise. Through manipulation of its form, it is shown that this filter can be achieved through appropriate choice of filter coefficients for a standard, second order phase lock loop. The validity of this design and of its underpinning assumptions and simplifications, is confirmed through both simulation and real signal experimentation. It is demonstrated that such a design offers improvement over traditional designs and shows potential as the basis for adaptive carrier phase tracking techniques.
1.3
Thesis Contributions
The primary contributions of this thesis can be categorised based on their nature. In effect, the thesis yields two tiers of contribution: theoretical analyses of some fundamental receiver functions are provided and, building upon these theoretical foundations and the insight they yield, novel design principles for each of these receiver functions are presented. The theoretical results include: a composite model for the signal-to-noise ratio degradation induced by filtering
and quantisation of the received signal, a mathematical model for the performance of frequency lock loops employing
differential-phase based carrier frequency discriminators and operating under weak-signal conditions, analytical, statistical characterisation of four differential-phase based carrier
frequency discriminators a mathematical model for the performance of the discrete-update phase lock
loop operating under weak-signal conditions, analytical, statistical characterisation of four carrier phase discriminators,
4
Section 1.3: Thesis Contributions a minimum mean square error (MMSE) optimal phase lock loop filter condi-
tioned on thermal noise and oscillator-induced phase noise. Employing these theoretical results in an optimisation of the receiver design procedure, the following design contributions were developed and presented: guidelines for the choice of filter bandwidth and centre frequency for one-, two-
and three-bit quantisation for the GPS L1 C/A signal, guidelines for the choice of filter bandwidth and centre frequency for one-bit
quantisation for the GIOVE-A B/C and GIOVE-B B/C signals, guidelines for the optimal choice of differential-phase based carrier frequency
discriminator for the steady-state operation of the frequency lock loop, design rules for the choice of carrier phase discriminator for the phase lock
loop operating in its linear region, guidelines for the choice of carrier phase discriminator for the phase lock loop
operating outside linear region, guidelines for the extension of the application of the optimal phase lock loop
to changing operation conditions. Theoretical results provided, and simplifying assumptions made, in this thesis have been extensively verified through Monte-Carlo simulation and, where practical, through experimentation with actual GPS L1 C/A signals.
5
6
Chapter 2
An Introduction to GNSS Signals and Receivers The GNSS systems considered in this thesis operate on the principle of time-ofarrival (TOA) ranging. Encoded in the special structure of these GNSS signals is the time that the signal was transmitted and information describing the satellite’s orbit and position. A receiver can estimate the range to each satellite based on the difference between the transmit time and the time of arrival of the signal. Geometrically, the user’s position can be ascertained given the range to three satellites, however, the TOA measurement is subject to one more degree of freedom; the receiver clock bias. Position determination, therefore, requires a minimum of four satellite observations, although many more observations can be utilized. To extract TOA information from the received signal both the carrier and the code modulation of the signal must be removed. A general discussion on the principles of TOA ranging can be found in, for example, [36, 61, 80]. The topic of signal reception and demodulation, focusing primarily on the carrier, is the focus of this thesis and will be introduced here. This chapter presents a detailed description of direct sequence spread spectrum (DSSS) code-division multiple access (CDMA) GNSS signals and the traditional receiver architecture and algorithms used to track them. The typical civilian propagation channel is discussed and its effect on the characteristics of the received signals are examined. Emphasis is placed upon the typical consumer grade receiver architecture and components thereof. The non-idealities of low-grade receivers are analysed and the resulting degradation in receiver performance is addressed. The primary design considerations and performance metrics of two key baseband receiver operations, namely, the FLL and the PLL are also studied. 7
Chapter 2: Introduction to Signals and Receivers
2.1
Overview of the GNSS signal structure
The focus of this thesis is the tracking of DSSS GNSS signals. Whilst the primary focus is the tracking of the GPS L1 C/A signal, GIOVE-A E1 B/C and GIOVE-B E1 B/C are also considered. Only these three specific signal examples are considered here and so, for brevity, the GPS L1 C/A signal will simply be referred to as the GPS signals while the GIOVE-A E1 B/C and GIOVE-A E1 B/C will be referred to as the GIOVE signals. As the GPS and GIOVE signals are quite similar, sharing the same centre frequency and spreading code chipping rate, a general description and model of this kind of GNSS signal will be presented here, discussing the aspects of the two systems which differ, as they arise. The composite signal incident on the receiver’s antenna, denoted here by rRF ptq, can be described as: rRF ptq
¸
P
siRF ptq
n pt q
(2.1)
i Ssig
siRF ptq
a
2Pi di ptq ci ptq sin pωRF t
θi ptqq ,
(2.2)
where Ssig is the set of satellite signals in view, visible as either direct line-of-sight (LOS) or as reflected signals, siRF ptq denotes the signal received from the ith satellite
and n ptq denotes the additive thermal noise. A simplified illustration of a binary
phase shift keyed (BPSK) DSSS modulated waveform is shown in Figure 2.1. The various parameters in (2.2) represent the following signal properties: Pi is the total signal power of the ith signal, in watts, incident on the antenna.
It is usually expressed in terms of dBW. ωRF is the nominal RF carrier frequency in units of rads1 . For the GPS L1
C/A, the GIOVE-A E1 B/C and the GIOVE-B E1 B/C signals this is 1575.42 MHz (or ωRF
2π 1575.42 106 rads1).
di ptq represents the bi-podal data signal, assuming values of t1, 1u. For the
GPS signals, di ptq has a bit rate of 50 bps. For the GIOVE data signals
(GIOVE-A E1 B and GIOVE-B E1 B), di ptq has a bit rate of 250 bps while
for the pilot GIOVE channels (GIOVE-A E1 C and GIOVE-B E1 C), which are not data modulated, di ptq is simply unity.
ci ptq is the signal spreading sequence. For the GPS signals, this sequence
is simply a non-return to zero (NRZ) pseudo-random noise (PRN) sequence, unique to each satellite, with a nominal rate of 1.023 Mcps. This sequence imparts upon the signal its DSSS CDMS characteristics. For the GIOVE signals, the spreading sequence is a composite signal, consisting of one or more sub carriers which are modulated with a PRN sequence. 8
Section 2.1: Overview of the GNSS signal structure
d(t)
c(t)
d(t)c(t) sin(ω t) RF
sRF(t)
Figure 2.1: Composition of a BPSK DSSS waveform such as GPS L1 C/A.
θi ptq is the phase process of the ith received signal. This is a composite signal
which contains both deterministic and stochastic effects. The components of this process include Doppler-induced phase variations, ionospheric and tropospheric effects and phase noise induced by the satellite local oscillator.
The thermal noise term, n ptq, is assumed to be additive white Gaussian noise
(AWGN) with a one-sided power spectral density (PSD) of N0 .
Central to the operation of GNSS are the correlation properties of the spreading code sequences. These sequences provide: spectral separation between the GNSS signals and other terrestrial signals; quasi-orthogonality between other satellite signals; interference rejection and anti-jamming capability; and, most importantly, facilitate TOA measurements. The spreading codes are generated from a sequence of values from the set {-1,1}.
The set of all sequences is chosen carefully to satisfy a number of criteria, namely, to ensure that each code: has one autocorrelation peak and that it has low autocorrelation values for
offsets greater than zero, has a low cross correlation with other codes in the set, is approximately zero mean.
The GPS signals use a set of spreading codes known as Gold codes [44, 57, 94]. Specifically, the codes used for the GPS L1 C/A signals are generated from the modulo-2 addition of two maximal length m-sequences and have length 210
1
1023 chips. These codes ensure a minimum autocorrelation main-peak to side-peak 9
Chapter 2: Introduction to Signals and Receivers ratio of 23.93 dB and also ensure a minimum autocorrelation main-peak to cross correlation ratio of 23.8 dB(1) [90]. One attractive feature of Gold codes, which contributed to their use in the GPS system, is the ease with which they can be generated. The codes used for the GPS signals are, generally, generated using tenbit linear feedback shift-registers (LFSRs) and, therefore, requite very little memory or computational resources, which was a significant consideration for early receivers. The GIOVE signals use a set of codes similar to those of the GPS signals, these codes are longer than those of the GPS signals, the GIOVE-A E1 B and GIOVE-B E1 B signals having length 4092 and the GIOVE-A E1 C and GIOVE-B E1 C signal having length 8184 [89]. The final signal specification for the Galileo OS signals, E1-A and E1-B, defines signals with spreading codes of length 4092, similar to those of the GIOVE-A signals. However, these signals, rather than being generated from LFSRs, are random codes which must be stored in the receiver. For this reason, they are referred to as memory codes [88]. The exact details of the codes used, and their lengths, is not the focus of this thesis, and contributes little to the work to follow. More significant is their materialisation in the modulated waveform. Each bit of the code sequence is modulated onto a short pulse, known as a chip. A series of these chips forms the spreading code. If the ith element of the code sequence, of length Ncode , is denoted ci , and
the short pulse, of duration Tc , is denoted chipptq, then the spreading code can be
expressed as:
8 ¸
cptq i
8
ci chippt iTc q,
(2.3)
where the function chipptq is nonzero only in the region 0
¤ t Tc
and is zero
elsewhere and the index, i, of the code bits, ci is assumed to be taken modulo Ncode (that is cNcode
1
c1 ). For the GPS L1 C/A signal, chipptq is simply a
rectangular pulse of unit amplitude and of duration Tc
1{1.023 106 s (i.e. the
reciprocal of the chipping rate). This is depicted in Figure 2.2 (a). The GIOVE-A E1 B/C signals use a chip materialisation known as binary offset carrier, specifically BOC(1,1) is employed [89, 61]. The BOC chip, often referred to as a sub-carrier, is a square wave of period equal to the chip duration. Such a chip is depicted in 2.2 (b). The GIOVE-B E1 B/C signal uses a more elaborate chip materialisation comprising two different sub-carriers. Known as composite binary offset carrier, CBOC is defined as the weighted sum of two square waves, producing a four-level waveform [89, 34, 4]. The GIOVE-B chip materialisation, in particular, is most significant, as it is the materialisation chosen for the final Galileo E1 B/C signals (1)
This value is accurate for simple cross-correlation of Gold codes. However, when signal Doppler is considered, the effective cross correlation value may increase, resulting in an effective minimum autocorrelation main-peal to side-peak ratio of approximately 17 dB and an autocorrelation mainpeak to cross correlation ratio of 21.6 dB [90]. A more thorough study of these cross-correlation effects, extended to consider the Galileo E1 OS signals is presented, for example, in [120, 48].
10
1
1
0.5
0.5 chip(t)
chip(t)
Section 2.1: Overview of the GNSS signal structure
0
0
−0.5
−0.5
−1
−1 0
0.2
0.4 0.6 t/Tc
0.8
1
0
1
1
0.5
0.5
0
−0.5
−1
−1 0.2
0.4 0.6 t/Tc
0.8
1
0.8
1
0
−0.5
0
0.4 0.6 t/Tc
(b) GIOVE-A E1 B/C
chip(t)
chip(t)
(a) GPS L1 C/A
0.2
0.8
1
0
(c) GIOVE-B E1 B and Galileo E1 B
0.2
0.4 0.6 t/Tc
(d) GIOVE-B E1 C and Galileo E1 C
Figure 2.2: Spreading code chip materialisation.
[88]. These chip materialisations are depicted in Figures 2.2 (c) and (d). As the codes are of finite length, the sequence cptq is periodic in time t, with pe-
riod Tcode
NcodeTc. The autocorrelation function of the spreading codes, Rcodepτ q,
can, therefore, be defined as:
Rcode pτ q
1
T» code
Tcode
cptqcpt τ qdt.
(2.4)
0
Substituting (2.3) into (2.4):
Rcode pτ q
1
8 ¸
T» code
Tcode 0
i
8 ¸
8 j 8
ci cj chippt iTc qchippt jTc τ qdt.
Nothing that the limits of integration are 0 and Tcode , that Tcode
(2.5)
NcodeTc and that
chipptq is zero outside the region 0 ¤ t Tc , the summation over i can be reduced, omitting the elements which do not contribute to the integral. Thus: 11
Chapter 2: Introduction to Signals and Receivers
Rcode pτ q
1
Ncode ¸ 1
Tcode
i 0
pi »1qTc
8 ¸ j
8
ci cj
chippt iTc qchippt jTc τ qdt.
(2.6)
iTc
This result can be simplified further by introducing the autocorrelation function of
the spreading code chip materialisation, Rchip pτ q and the autocorrelation function
of the code sequence as:
Rchip pτ q
Cj
»Tc
chipptqchippt τ qdt
0 N¸ code
(2.7)
ci cij .
(2.8)
i 1
Substituting (2.7) and (2.8) into (2.6), and recalling that ci is periodic in Ncode , it can be shown that: Rcode pτ q
1
Ncode ¸ 1
Ncode
i 0
8 ¸
j
8
8 ¸ j
8
Cj Rchip pτ
ci cj Rchip pτ
pj iqTcq
jTc q.
(2.9)
The autocorrelation function, Rchip pτ q, is shown in Figure 2.3 for each of the
chip materialisations depicted in Figure 2.2. Equation (2.9) describes the generalised autocorrelation function of the spreading code defined in (2.3). It is periodic in τ
NcodeTc and its shape around its main autocorrelation peaks Tcode, 0, Tcode, 2Tcode, ... etc.) is defined by the autocorrelation
with period Tcode (centred at τ
function of the chip materialisation. Likewise, the autocorrelation side-peaks will have a shape defined by Rchip pτ q, albeit attenuated heavily by the coefficient Cj , which, as discussed previously, is, by design, very small for j
0. A simple illustra-
tion of Rcode pτ q for each of the four chip materialisations considered here is shown
in Figure 2.4. Note that the relative magnitudes of the coefficients Cj are not to
0, are exaggerated for clarity. The exact properties of cptq have a significant impact on the performance of the
scale and the values of Cj , j
system, influencing both the occupied spectrum of the RF signal [89, 88, 4] and the acquisition and tracking capability of receivers. Furthermore, as will be shown in
Chapter 3, Rcode pτ q has an impact on the processing losses of the receiver. A further
explanation of these processing losses will be presented in Section 2.4.
Whilst this section has aimed to present a detailed model of the GNSS signal, some simplifying assumptions have been made. Namely, the details of the constant 12
Section 2.2: Propagation Effects
1
Rchip(τ)
0.5
0
−0.5 −1
−0.8
GPS L1 C/A
−0.6
−0.4
−0.2
0 τ/Tc
GIOVE−A E1 B
0.2
0.4
0.6
GIOVE−B E1 B
0.8
1
GIOVE−B E1 C
Figure 2.3: Autocorrelation function of chip materialisations for the GPS L1 C/A, GIOVE-A E1 B, GIOVE-B E1 B and GIOVE-B E1 C chip materialisations.
envelope modulation which is applied to the GIOVE signals, have been omitted. A detailed explanation of this modulation scheme, known as coherent adaptive subcarrier modulation (CASM) can be found in, for example, [89, 88, 18, 4, 48]. Details of the tiered codes applied to the GIOVE-B E1 C and Galileo E1 C signals, omitted here, can also be found in these references. The motivation for neglecting these details will be clarified in Section 2.3 and further in Chapter 3. Essentially, as the signal processing techniques considered in this thesis are applied to each code in isolation, and over sufficiently short durations, the presence, or otherwise, of the various other components of the CASM signal can, reasonably, be neglected. Interestingly, this modulation scheme has not been adopted for the final Galileo signal specification. Instead a purely additive technique, known as modified interplex modulation, is being employed.
2.2
Propagation Effects
Propagation of the transmitted satellite signal from an orbital altitude of 20,000 km to 24,000 km to an earth-based user can incur significant signal degradation. Amongst these effects are signal attenuation, distortion and dilation. Some of these effects, and their manifestation in the received signal parameters of (2.2), are discussed here. 13
Chapter 2: Introduction to Signals and Receivers
Rcode(τ)
1 0.5 0 −0.5 −1
0
1
2
3
τ/Tc
4
5
6
7
8
5
6
7
8
5
6
7
8
5
6
7
8
5
6
7
8
(a) Code only
Rcode(τ)
1 0.5 0 −0.5 −1
0
1
2
3
τ/Tc
4
(b) GPS L1 C/A
Rcode(τ)
1 0.5 0 −0.5 −1
0
1
2
3
τ/Tc
4
(c) GIOVE-A E1 B
Rcode(τ)
1 0.5 0 −0.5 −1
0
1
2
3
τ/Tc
4
(d) GIOVE-B E1 B
Rcode(τ)
1 0.5 0 −0.5 −1
0
1
2
3
τ/Tc
4
(e) GIOVE-B E1 C Figure 2.4: Autocorrelation function of the spreading code and composite spreading code for each materialisation.
14
Section 2.2: Propagation Effects Spec. Ant. Gain (dBic) GPS L1 C/A GIOVE-A E1 B GIOVE-A E1 C GIOVE-A E1 B GIOVE-A E1 C Galileo E1 OS B Galileo E1 OS C
-160.0 -162.3 -162.3 -163.6 -163.6 -160.0 -160.0
Low Elev. 5 - 30 -10.0 -170.0 -172.3 -172.3 -173.6 -173.6 -170.0 -170.0
Moderate Elev. 30 - 60 -8.0 -168.0 -170.3 -170.3 -171.6 -171.6 -168.0 -168.0
High Elev. 60 - 90 -5.0 -165.0 -167.3 -167.3 -168.6 -168.6 -165.0 -165.0
Table 2.1: Minimum received power levels for a typical consumer grade receiver. The range of typical antenna gains assume an embedded miniature right-hand circular polarised (RHCP) antenna. The column entitled ‘Spec.’ indicates the specified minimum received signal power (MRSP) assuming a matched RHCP isotropic 0 dBic antenna as per the system interface control document (ICD) [57, 89, 88]. The maximum received power levels for the GIOVE and Galileo signals is specified as no more than 3 dB above the corresponding MRSP, while for the GPS L1 C/A signal an upper bound of 7 dB is specified.
2.2.1
Received Signal Power
A number of factors influence the signal power, Pi , available to the GNSS receiver. Whilst the transmitted power at the satellite is approximately 27 W [80], the power incident on the antenna of an earth based receiver, even for a LOS signal, will be of the order of 0.1 fW (-160 dBW). The factors which dictate this power loss include the satellite antenna gain, the free space propagation loss and the gain of the receiver’s antenna. The gain of the satellite’s antenna is a measure of the ability of the antenna to focus the transmitted power towards the earth. In the case of GPS L1 C/A, the antenna gain is approximately 13 dB. As the signal propagates through space, towards the earth, it disperses. The further the range from the transmitting antenna, the lower the power spacial density will be at the receiver (in fact the relationship is an inverse square one). This power loss is known as a path loss and, for the GPS and Galileo orbits, is in the region of 182 dB to 185 dB [80]. To convert this radiated power spacial density into a usable voltage within the receiver, a suitable antenna must be employed. Typical consumer grade GNSS receivers are equipped with miniature antennas, for example [100, 125], which are attractive, owing to their small outline and low cost. In terms of gain, these devices perform poorly, however, providing gains ranging from -5 dBic to 4 dBic, depending on the elevation of the satellite relative to the antenna. The result is a received signal power available to the GNSS receiver at the output of the antenna of approximately -160 dBW. A list of the typical minimum received signal powers, for a selection of antenna gains and elevations, is presented in Table 2.1. A thorough analysis of this propagation loss and the influence of the antenna on this MRSP is given in, for example, [80, 90]. For a typical civilian user, the figures of Table 2.1, as they pertain to the un15
Chapter 2: Introduction to Signals and Receivers obstructed LOS signal, present the best case scenario. Generally, civilian users are located in urban or indoor environments which can severely degrade the received signal quality. The indoor and urban canyon environments are characterised by significant signal attenuation and the presence of reflected signals. LOS signals in these environments, incident on a receiver’s antenna, are likely to have propagated through one or more obstructions. Common building materials, such as wood, steel, brick and concrete can significantly attenuate the signal. Generally, a certain portion of the incident signal will be reflected away from the obstructing material and the remainder will be further attenuated and refracted through the material. The amount by which the LOS signal is attenuated depends on the exact material causing the obstruction. This varies greatly and, accordingly, the attenuation which is observed varies. A thorough analysis of the propagation losses incurred by a variety of building materials is presented in [105], wherein it is illustrated that a signal, similar to the L1 or E1 GNSS signals, will experience attenuation levels ranging from 5 dB to 35 dB when propagating through various materials. A selection of these results is collated in Table 2.2. Of course, whilst this information can be used to calculate the attenuation caused by a particular propagation path, more useful is a picture of the typical received signal power over a range of satellite elevations and propagation paths. For example, [35] presents a collection of measured signal power estimates for the unobstructed LOS environment, for an antenna obstructed by foliage, and for an indoor environment. In particular, for the indoor environment, it is shown that the received signal power can be attenuated by an amount ranging from 1 dB to as much as 40 dB, for satellites of all elevation. Similar results and conclusions are presented by, for example, [33, 68].
2.2.2
Multipath
Unfortunately, the propagation of the GNSS signal through urban and indoor environments does not simply incur an attenuation of the signal power. Generally, the received GNSS signal includes a LOS signal and a number of non-line-of-sight (NLOS) signals. These signals are ones which have been reflected off nearby obstacles one or more times before reaching the antenna. An example of the propagation of a signal in an urban environment is illustrated in Figure 2.5. A reflected signal’s arrival at the receiver’s antenna will be delayed relative to the LOS signal. Reflection of the signal will also induce a power attenuation and a shift in the carrier phase. The signal at the antenna can, thus, be modelled as a linear combination of attenuated and delayed versions of the LOS signal. Indeed, this type of model is common and numerous models of the multipath environment have been presented (see, for example, [56, 55, 111, 94]). Whilst multipath effects can significantly degrade the accuracy of a receiver’s 16
Section 2.2: Propagation Effects
Material Glass
Lumber
Plywood Brick
Brick-Faced Concrete Brick-Faced Block Plain Concrete
Masonry Block Reinforced Concrete
Thichness (mm) 6 13 19 38 76 114 19 90 180 270 90 / 102 90 / 203 90 / 194 102 203 306 203 406 203
Attenuation (dB) 1.2 2.8 3.6 3.5 4.2 5.8 2.5 4.5 6.5 9.0 16.5 31.0 10.5 14.0 26.0 34.5 11.0 17.5 33.0
Table 2.2: Typical signal attenuation caused by propagation through various building materials. All data sourced from [105].
NLOS
LOS
Antenna
Figure 2.5: Multipath propagation of GNSS signals.
17
Chapter 2: Introduction to Signals and Receivers position estimates (through biasing the pseudorange measurements) of interest in this work is the impact on the carrier loop. In particular, one multipath effect which poses difficulties to carrier tracking algorithms is that of fading. When two or more versions of the GNSS signal (LOS and NLOS) are incident on the receiver’s antenna, the ensemble can constructively combine to increase the apparent received power, or they can destructively combine to reduce the apparent received signal power, depending on the relative carrier phase of the signals. This effect, known as fading, can significantly reduce the effective received signal power for GNSS receivers operating in urban or indoor environments, a discussion of these issues is given in, for example, [80, 94].
2.2.3
Thermal Noise
As indicated earlier, the thermal noise term, n ptq, of (2.2), is assumed to be AWGN
with a one-sided PSD of N0
kB TE .
The constant, kB , is Boltzmann’s constant
and TE is the effective receiver noise temperature. Note that TE is the effective temperature in Kelvin and must incorporate the antenna, amplifier and cable losses via Friis’ formula [80, 114]. For typical receivers, depending on the noise figure of the antenna, cabling and LNA employed, the value of N0 is typically in the range
201 dBW/Hz to 203.8 dBW/Hz.
The result of these propagation and receiver effects is that the GNSS signal received by a civilian receiver in an urban or indoor environment can be observed with a carrier-power-to-noise-floor-power ratio (often abbreviated to carrier-to-noise ratio or to CNR and denoted here by C {N0 ) is typically in the range 48 dB Hz to 15 dB Hz.
2.2.4
Interference
An undesired RF signal which is incident on the receiver’s antenna is considered an interference. Undesired signals are those that are not belonging to the set of signals that the receiver intends to process. Interfering signals can have a serious impact on the performance of GNSS receivers, either reducing the accuracy of the receiver or rendering it entirely inoperable. Not only are receivers susceptible to in-band signal sources, out-of-band signal sources can also pose an interference threat. As the received GNSS signal is weak, strong out-of band sources can propagate through the receiver’s antenna and filters with strength comparable to that of the GNSS signals [16, 61]. An interfering signal can arise from both intentional and unintentional sources. Examples of intentional interferences include: signal jamming, where a malicious source emits a signal intended to overwhelm a particular set of received signals; or spoofing, where the source emits one or more replica GNSS signals intended to provoke the receiver to estimate a false position [61, 59]. Unintentional interferences 18
Section 2.2: Propagation Effects generally originate from out-of-band sources. Signals which occupy nearby signal bands (such as local area network (LAN) or radar [80], for example) can result in interfering signals within the GNSS band, either in the form of intermodulation products or in the from of harmonics of the original signal, and are manifest as either spontaneous, burst emissions or continuous wave (CW) signals. Even if a particular emission resides outside the GNSS band, depending on its strength relative to the GNSS signal, it can impair the receiver front end performance, thereby rendering it an interference. Interfering signals influence a number of receiver components including: the front-end; the analogue-to-digital converter (ADC); and the baseband functions. As the MRSP of GNSS signals is low, the receiver will feature a high gain LNA (with a gain of the order of 100 dB). Strong interfering signals can drive the LNA into saturation, thereby obliterating the presence of the received GNSS signals. The nonlinearity of the ADC is also a weakness with respect to interference. The presence of a CW interfering signal can significantly increase the quantisation loss experienced by the receiver, relative to the interference-free loss [14, 61, 59]. Finally, the presence of an interfering signal distorts the statistics of the correlator values (as discussed in the next section). Generally, the effect of the interfering signal is modelled as a reduction in the received signal to noise ratio, either as a signal-to-noise-andinterference ratio (SNIR) [14], or as some form of effective carrier to noise ratio [61]. The exact performance degradation is typically a function of the specific details of the signal, the interference and the baseband process (see, for example, [17, 59, 61]).
2.2.5
Doppler and Doppler Drift
Other than the effects of propagation on the received signal power, propagation from a satellite orbit to an earth-based user also influences the observed carrier frequency. The satellite traverses its orbit with a period of 11 hours 58 minutes while the earth-based user traverses a circular path around the earth’s axis with a period of 24 hours. Using a simple model of the relative satellite-to-user displacement, the relative satellite-to-user elevation, relative velocity and relative acceleration can be inferred. The effect of the relative satellite-to-user velocity, v, can be related to Doppler frequency, ωd , via [61, 80, 94]: ωd
ωRF Cv ,
(2.10)
where ωRF is the GNSS RF carrier frequency and C is the speed of light. Note that the right hand side of (2.10) is negative, implying that when the satellite is approaching the receiver, an increase in the carrier frequency will be observed. In a similar fashion, the rate of change of Doppler frequency, often referred to as Doppler drift, can also be computed. A simple illustration of the typical Doppler and Doppler drift experienced by 19
Elevation (deg)
Chapter 2: Introduction to Signals and Receivers
50 0 −50
Doppler (Hz)
0
Visible
Visible
3
6
9
12 15 Time (hrs)
18
21
24
3
6
9
12 15 Time (hrs)
18
21
24
3
6
9
12 15 Time (hrs)
18
21
24
5000 0 −5000 0
Drift (Hz/s)
1 0 −1 0
Figure 2.6: Variation of Elevation, Doppler frequency and Doppler drift with time for a stationary user situated at 53 North of the equator.
a receiver is shown in Figure 2.6, wherein the curves are calculated assuming a simplified circular orbit and a stationary user situated at 53 North of the equator.
Of course, the satellite is visible only when its elevation is positive (in fact, typically, only satellites with elevation above 5 are usable). This coarse, simplified example suggests that for the duration of time that the satellite is visible, the user will experience Doppler frequencies, which are always decreasing, and are in the region of 5000 Hz; whilst values of Doppler drift for these periods range from 0 to -1 Hz/s. This effect forms the deterministic component of the ith satellite phase process, θi ptq.
2.2.6
Atmospheric Effects
Propagation through the earth’s atmosphere can also have a negative effect on the quality of the received GNSS signal. Early estimates attributed a constant power loss of up to 2 dB to the atmosphere [80, 57] although recent measurement campaigns have resulted in a revision of this estimate down to approximately 0.5 dB. The characteristics of the ionosphere and the troposphere can have a further impact on the received signal. The troposphere extends from sea level to anything from 9 to 16 km kilometres above sea level and contains most of the mass of the earth’s 20
Section 2.2: Propagation Effects atmosphere, consisting mostly of water vapour, nitrogen and oxygen [80]. One primary effect of the troposphere is its effect on the propagation speed. Depending on the elevation of the satellite and the time of day, the troposphere can induce a delay in the transmitted signal. This delay, when expressed in meters, can range from a minimum of 2 m and, for low elevation satellites, can exceed 15 m [80, 58]. The properties of the troposphere are relatively stable and change little over time and, therefore, can be well modelled, facilitating the removal of these troposphereinduced errors. Typical troposphere models can provide modelling errors of the order of tens of centimetres [80, 58]. A further effect of the troposphere is that of tropospheric scintillation, which is caused by turbulence in the lower region of the troposphere. The effect of this scintillation on the received GNSS signal is that of a randomly varying received signal power. This effect results, typically, in an rms scintillation of 0.3 dB for high elevations and 0.9 dB for low elevations. A detailed description of the some tropospheric scintillation models and typical values is given in [58], for example. Whilst the troposphere does indeed have a significant impact on the GNSS signals, from a position determination perspective, the effects on the signal are relatively small and slowly varying and, so, are easily accommodated by typical carrier tracking algorithms. These effects must, of course, be borne in mind, but need not be the focus of receiver design routines. They are, therefore, in the context of this thesis, neglected. Added to the effects of the troposphere, the ionosphere, which extends from a height of approximately 50 km to above 1000 km above sea level, has a significant impact on the properties of the received GNSS signal. The physical properties of the ionosphere change between day and night, from day to day and seasonally [80, 67]. Comprised of ionized gasses, the speed of propagation of GNSS signals through the ionosphere is a function of the total electron content in the signal path. The effective propagation delay induced by the ionosphere is significantly greater than that of the troposphere, ranging from several meters to tens of meters and also varies more rapidly than the troposphere. Being a dispersive medium, the ionospheric delay differs for different frequency carriers, and so can be readily estimated by multifrequency receivers (e.g. L1 and L2 for the GPS system). For civilian applications, however, this is, generally, not necessary, and is only used for precise positioning applications. While the properties of the ionosphere vary rapidly as compared to the troposphere, the net effects on the received carrier phase are relatively slow, and are easily accommodated by typical carrier tracking loops. The ionosphere does, however, pose one significant threat; in certain regions around the earth’s equator, where the ionosphere is most active, irregularities in the depth of the ionosphere and in the total electron content can induce large, sudden signal fading and random phase fluctuations. These fading effects can be relatively large, exceeding 15 dB in some cases [67, 126] and can induce simultaneous random 21
Chapter 2: Introduction to Signals and Receivers phase variations. The combination of these effects can stress narrowband receivers and to the point of loss of lock. Outside of these regions of high ionosphere activity, fortunately, the ionosphere poses little threat to typical GNSS receivers [126].
2.3
The Matched Filter
Estimation of the various parameters of the received GNSS signal is central to the process of position determination and so it is unsurprising that the fundamental operation of the GNSS receiver is that of a matched filter. Matched filters arise in the the GNSS receiver in many guises, as implementations of the square-law signal detector in the receiver’s acquisition algorithms [80, 61, 110], as the maximum likelihood phase estimator, central to the arctangent-based PLL [94], and as the optimum amplitude estimate used to provide early, prompt and late power estimates to the receiver’s delay lock loop (DLL) code discriminators. The properties of the matched filter and its manifestation in typical GNSS receivers is presented here. Depicted in Figure 2.7 is a block diagram of the typical GNSS correlator. Locally generated replicas of the code and carrier are multiplied by the intermediate
frequency (IF) signal, rRF ptq. The product is then integrated, or accumulated, over a period, TI , known as the integration period (or the correlation period). The exact
order of code and carrier multiplication is, of course, unimportant. Multiplication by the local carrier signals reduces the signal to baseband, or approximately so, while the process of code multiplication despreads the desired signal and is known as ‘code-wipe-off’. The integration process is often approximated in modern digital receivers by a summation, or accumulation. In terms of receiver operation, for band-limited sampled signals, the two processes are approximately equivalent. The result is the correlator pair, Im and Qm , where the index m denotes the mth correlation period: mTI
¤ t pm
1qTI . This implementation provides the maximum
signal to noise ratio for the values Im and Qm [94]. The properties of Im and Qm
are crucial to the performance of subsequent receiver operations and have been well documented. It can readily be shown that, for received signals of the form (2.2), Im and Qm are well approximated by [61, 80, 27, 28, 114]:
Im Qm
?
P sinc δω2cTI Rcode pδτ q cos pδθq
? δωc TI P sinc 2 Rcode pδτ q sin pδθq
ni
(2.11)
nq ,
(2.12)
where δτ , δθ and δωc represent the mean code phase, mean carrier frequency and mean carrier phase error over the correlation period and sinc pxq
sin pxq {x.
lation period and influences the angle of the complex value Im
jQm . This is the
The
carrier phase error, δθ, represents the constant mean error observed over the corre-
22
Section 2.4: Receiver Non-Idealities and Associated Losses
Integrate & Dump
Im
Integrate & Dump
Qm
2 sin(ω IF t )
rRF (t )
2 cos(ω IF t )
From Front End
c(t )
Figure 2.7: Block diagram of a typical correlator implementation in a GNSS receiver
error which is tracked by the PLL. The carrier frequency error, δωc , represents the net difference in frequency between the receiver local carrier frequency and the sum of the nominal IF frequency, the Doppler-induced frequency and satellite oscillatorinduced frequency errors. The terms ni and nq represent the noise terms which corrupt the correlator output values. They are typically modelled as zero mean i.i.d AWGN terms with variance given by(2) [61, 80, 27, 28, 114]: Var rni s Var rnq s
N0 . 2TI
(2.13)
Equations (2.11) and (2.12) represent the, rather simplified but widely accepted, model of the correlator outputs. It is from these two metrics that all of the receiver signal estimates and user position, velocity and time (PVT) estimates are derived.
2.4
Receiver Non-Idealities and Associated Losses
The ideal model for the correlator values, presented in Section 2.3, neglects a number of receiver non-idealities which result in a degradation of the quality of Im and Qm . Receiver components inducing these losses include the front-end filter, the analogue to digital converter, or quantiser. An introductory description of these components and their associated losses is presented here.
2.4.1
Filtering Losses
GNSS signals are transmitted at high radio frequency (RF) frequency and, owing to their DSSS implementation, are quite wideband. To facilitate down-conversion to IF and, subsequently, to baseband and to prevent excessive aliasing when sampled, (2)
Note that some authors will choose to normalise the correlator values differently. For example, such that the correlator values have unity power or such that the noise terms have unity variance or such that the correlator values have unit amplitude. For any particular normalisation, the signal and noise parts will be scaled by the same amount and the signal-to-noise ratio will remain unchanged.
23
Chapter 2: Introduction to Signals and Receivers
GPS L1 C/A
GIOVE−A E1 B
GIOVE−B E1 B
GIOVE−B E1 C
−30 −35
PSD (dBW/Hz)
−40 −45 −50 −55 −60 −4
−3
−2 −1 0 1 2 Offset from carrier frequency (MHz)
3
4
Figure 2.8: Normalised one-sided PSD of the received GNSS signals versus offset from the carrier frequency.
it is necessary to band-limit the received RF signal. This band-limiting is usually implemented in numerous stages throughout the down-conversion process [40, 114, 61], but the filtering effect can, generally, be modelled as a single bandpass filter [28, 27, 11, 50, 80]. The primary negative effect of the front-end filter is that it removes, or attenuates, signal power which lies outside its passband. The result is a reduced signal power at the output of the correlator. This effect is best envisaged by examining the PSD of the transmitted signals. The PSD of the received signals will simply be that of the spreading code, shifted such that it is centred at the carrier frequency. It can readily be related to the spreading code autocorrelation function via the Wiener-Khinchin theorem [26]:
8 ¸
Scode pω q Ncode k
8
Ts Rcode pkTs q ej2kω ,
(2.14)
where Scode pω q denotes the two-sided PSD of the spreading code. The normalised
PSD of the four GNSS signals presented in Section 2.1 is shown in Figure 2.8.
For optimum performance, it is clear that a front-end filter centred at the received signal carrier frequency should be used and that the wider the filter bandwidth, the more signal will be admitted and the higher the signal-to-noise ratio will be. Interestingly, for the GPS L1 C/A signal, approximately 90% of the signal power is contained within the main lobe and so a 2 MHz front-end filter will incurr approximately a 0.5 dB loss [114, 61]. For the GIOVE E1 and Galileo signals, however, 24
Section 2.4: Receiver Non-Idealities and Associated Losses
10 GPS L1 C/A GIOVE−A E1 B GIOVE−B E1 B GIOVE−B E1 C
8
Lfilt (dB)
6
4
2
0 2
4
6
8
10 12 Bf (MHz)
14
16
18
20
Figure 2.9: Front-end filter loss versus one-sided front-end filter bandwidth.
there are two main lobes, and to achieve a comparable loss, a 4 MHz front-end filter is required [61]. A simplified loss model, commonly employed [13, 14, 114, 52, 61], is that the loss incurred by the front-end filter is equivalent to the signal power that it eliminated. A loss coefficient is generally defined as:
Lf ilt
2πB » f
2πBf
Scode pω q dω,
(2.15)
where Bf is the one-sided, equivalent low-pass filter bandwidth in Hz. Using this model, a plot of Lf ilt versus Bf is shown in Figure 2.9. Whilst this model accurately estimates the signal power available to the correlator, this is not, necessarily, proportional to the signal power available in the terms Im and Qm . In particular, the filter will have an impact on the phase of the received signal, causing some frequencies to be delayed more than others. Ideally, a matched filter would compensate for this effect by applying a complimentary filter to the local replica signal. In practice, however, this is not usually done, and the result is that the power in the values Im and Qm is less than the total received power. Nevertheless, the use of this model is prevalent in the literature (see, for example, [50, 13, 14, 11, 114, 52, 61]). Note that this further loss is not accounted for in this model. A discussion of this effect and a revised loss model is presented in Chapter 3. 25
Chapter 2: Introduction to Signals and Receivers A slightly more general treatment is presented in [50], wherein the filter is not assumed to be a brick-wall filter. This approach may, indeed, provide more accurate results, but, two further issues have been neglected. Firstly, an equivalent low-pass filter is assumed. For a linear system this is a reasonable. However, modern GNSS receivers are, invariably, digital receivers and employ an ADC. The non-linearity of the ADC renders all assumptions of linearity invalid. It will be shown in Chapter 3 this this non-linearity can have a stark impact on the resultant loss. Secondly, it is assumed that the front-end filter has a negligible impact on the PSD of the noise, stating that the front-end filter has a far wider pass-band than the correlator. While this is true, the effect of the local replica spreading code has been neglected. The local code replica, in fact, has a significant impact on the PSD of the thermal noise and a corresponding impact on the total loss. In subsequent work [51], although the effects of filtering on the noise components is addressed, the effect of the code wipe-off is still neglected and, as shown in Chapter 3, can lead to an underestimation of the correlator signal-to-noise ratio. It will be shown in Chapter 3 that the loss induced by the front-end filter cannot be related to Bf alone. It will be shown that, in fact, the total loss incurred is a function of Bf , the filter centre frequency, the autocorrelation of the spreading code, and the properties of the AGC quantiser. A comprehensive loss model will be shown, which, interestingly, leads to an optimum front-end design which contradicts that which would be derived from the loss model of Figrue 2.9. In particular, it will be shown that an increase of Bf does not, necessarily, increase correlator signal-to-noise ratio.
2.4.2
Quantisation Losses
Modern GNSS receivers are almost invariably digital receivers, being either dedicated digital hardware receivers or, more recently, hybrid hardware/software receivers [40, 3]. Typically, an analogue front-end is employed which down-converts the RF signal to IF at which point the signal is sampled and quantised to produce a discrete-time and discrete-amplitude signal. Given the recent increase in the achievable sample rate of ADCs, a process known as direct-digital down-conversion has become feasible [49, 107, 95]. Direct-conversion receivers apply a bandpass filter to the RF signal and then directly sample the filtered signal, aliasing the signal to a desired centre frequency [117]. In either case, the analogue signal must be rendered to one of a finite number of levels, usually evenly spaced and centred around zero volts. Typically, such receivers use a low resolution quantiser (e.g. one, two or three bits) to render the analogue signal. This coarse quantisation induces considerable errors in the instantaneous amplitude of the digital signal, relative to the analogue signal. The resulting degradation in the signal-to-noise ratio of the Im and Qm values is often termed the correlation loss, or quantisation loss. An example of this quantisation 26
Section 2.4: Receiver Non-Idealities and Associated Losses
5
Amplitude
3 1 0 −1 −3 Analogue −5 0
0.1
0.2
0.3
0.4
1−Bit 0.5 Time
0.6
2−Bit 0.7
0.8
0.9
1
Figure 2.10: Example of one- and two-bit quantisation of an analogue signal.
operation is depicted in Figure 2.10 for one- and two-bit quantisation. The impact of this loss on the performance of a receiver’s DMF has been of interest for many years (see, for example, [71, 24, 14, 114, 12, 63, 16]). Early work on the optimisation of the quantiser began, interestingly, with the problem of quantising analogue input signals such that they could be transmitted in digital form [77], as opposed to the GNSS case, where analogue signals are transmitted and quantised upon reception. While this problem differs somewhat from the quantisation problem considered here, the model used was that of a Gaussian distributed input signal, which is comparable to the GNSS signal. A mathematical framework which formally justified the symmetric quantiser configuration was introduced in [77]. Following from this work, further optimisation criteria which sought to maximise the detection performance of a quantised signal, albeit with a non-uniform quantiser, were developed in [63, 64]. An early example of a loss analysis of a system which would be more comparable to a GNSS receiver is that of [71], wherein a thorough treatment of the DMF in the presence of multi-bit quantisation is presented. Although it provides insight into the losses associated with quantisation in a DMF, analytical results, with which a receiver optimisation could be performed, are lacking and the investigation considers only the white noise case and a limited range of receiver parameters. One popular, early, study of the effects of band-limiting, sampling and quantisation on the DMF is that of [24], which considers the losses experienced by a pulse-code modulation (PCM) system. Using an idealised brick-wall filter, an analytical development of the band-limiting losses are presented, followed by thorough 27
Chapter 2: Introduction to Signals and Receivers analysis of the associated quantisation losses, in terms of the signal-to-noise ratio of the DMF. Moreover, the work illustrates the importance of the optimisation of the quantisation threshold, based on spectral density of the corrupting noise, a tactic widely employed in current GNSS receivers. It also proposes a reasonable relationship between filter bandwidth and symbol rate. It must be stressed, however, that the work in [24], has been developed around the principles of PCM transmission, which bears only a vague resemblance to the GNSS signal format. For example, it assumes synchronous sampling, a feature which is impractical in GNSS receivers, as they access multiple asynchronous channels from the same sampled sequence. The signal assumed has no spreading code, which, as will be shown in Chapter 3, has a significant impact on the resultant loss, due to the code wipe-off process. Finally, similar to the cases of Section 2.4.1, it assumes an idealised filter. In particular, it will be shown that, owing to the nonlinear nature of quantisation, one cannot always represent a bandpass filter with its equivalent low-pass filter, again, this will be illustrated in Chapter 3. The assumption that the processing losses experienced by a PCM receiver are comparable to those experienced by a GNSS receiver is widely applied in the GNSS fields, for example in [114, 12, 20, 5, 92]. These results are not, however, comparable and, in fact, can lead to an overestimation of the total processing loss. In the context of GNSS, recent simulation-based work has examined these effects in direct sequence spread spectrum (DSSS) signal reception for a variety of modulation types [14]. This work was subsequently extended to consider the multi-bit quantisation case [12]. Another simulation-based investigation of this loss is presented in [92], which considers the effect of both filtering and of quantisation on the correlation loss for a variety of different GNSS modulation schemes. These investigations present tabulated loss estimated for a variety of quantiser configurations. Also, a comprehensive theoretical analysis of the effects of quantisation alone (neglecting the front-end filter) has been developed in [16] which has laid the foundation for the work presented in Chapter 3. The primary results of Chapter 3 illustrate that, when considering the correlation losses incurred by the front-end filter and the quantiser, neither component can be considered in isolation. It is shown that both components influence the correlation function of the corrupting noise, which, in turn, impacts upon the correlator signalto-noise ratio. It will be shown that, in fact, the processing loss is a function, not only of the filter bandwidth and quantiser levels, but also of the sample frequency, and filter centre frequency and modulation type. To account for these factors, an analytical loss model is presented with which novel front-end optimisation can be performed. 28
Section 2.4: Receiver Non-Idealities and Associated Losses
2.4.3
The Local Oscillator
At the heart of any radio telecommunication device is some form of reference frequency source. Owing to the nature of CDMA TOA positioning systems, reference frequency sources, perhaps, play a more significant role in GNSS devices than in most other modern telecommunication devices
(3) .
Both the accuracy and stability
of these frequency sources are key factors in the achievable accuracy and robustness of the GNSS receiver. Real frequency sources, in the form of oscillators and resonators are, of course, not perfect and can be the limiting factor for consumer grade receivers. For example, they can impose an upper limit on the coherent integration period [97, 41], can impose a lower limit on the PLL filter bandwidth [29, 80] and can limit the range of applications in the case where oscillators are particularly sensitive to high dynamics and vibrations [39, 66, 118]. A thorough understanding of the oscillator’s characteristics can facilitate the design of GNSS receivers which can accommodate these imperfections. Oscillators are, generally, characterised in terms of their time process. The Oscillator Model The sinusoid emitted by a reference oscillator can be modelled by [79, 73]:
OSC ptq A ptq sin pΨ ptqq Ψ ptq 2π pf0
δf0 q t
(2.16) N ¸ 2πδfi1
i 2
i!
ti
ψ pt q,
(2.17)
where A ptq represents a time varying amplitude and Ψ ptq represents the phase process of the oscillator. The phase process comprises of the deterministic effects, f0
and δfi1 , the nominal frequency and the ith frequency drift, respectively, and a ran-
dom process, ψ ptq. The deterministic drift components are often termed long term
instabilities, as their influence on the phase process grows with time. In contrast, the random phase process, ψ ptq is referred to as a short term instability as its influence
becomes dominant when the phase process is observed over short periods of time. As oscillators can be designed to exhibit a variety of different nominal frequencies and are, generally, used to drive different frequency synthesizers(4) , it is convenient to define a common, normalised, phase process with which different oscillators can (3)
Features of CDMA TOA positioning systems which impart this sensitivity, for example, include: the high RF carrier frequency, which serves to amplify the instability of the oscillator; the very fact that the position solution is derived from time-based measurement; a low received power, which necessitates long coherent integration times; and the fact that, generally, the receiver must be a portable device and, therefore, must exhibit a small outline and a relatively low power consumption (two features which are rarely associated with precision oscillators). (4) Also, as the oscillator sinusoid is digitized or saturated to a square wave, the amplitude process, A ptq, can, reasonably, be neglected.
29
Chapter 2: Introduction to Signals and Receivers be compared. An oscillator’s time process, T ptq, is defined as: T ptq
1 Ψ ptq 2πf0
t
δf0 t f0
(2.18) N ¸ δfi1
i 2
i!f0
ti
1 ψ ptq. 2πf0
(2.19)
Noting that, by definition, ψ ptq is a zero mean process, it is clear that the de-
terministic instabilities must be reasonably small, in order that the oscillator be a usable frequency reference. In particular, higher order frequency drifts must be small. Considering oscillators in typical GNSS receivers, this is generally achieved, for example, the drift rate (change in
δf0 f0
per unit time) of a quartz oscillator is
typically less than one part in ten million, per year [79]. Oscillators with acceptable frequency offset and drift rates are readily available and, in terms of receiver performance, these deterministic instabilities can reasonably be neglected. The same
is not true for ψ ptq. Frequency references with negligible random instabilities are notoriously uncommon and are generally large, expensive devices that consume a significant amount of power. Hence, random instabilities must be estimated and modelled and receivers must be designed to accommodate them. Phase Noise Measurements Characterising ψ ptq, unfortunately, is not an easy task. To capture the properties
of ψ ptq alone, a metric is required which is insensitive to the deterministic oscillator instabilities. Such metrics can be devised by examining the evolution of T ptq. The first increment of T ptq, denoted ∆1 T pt; τ q, is defined as [79, 73]: ∆1 T pt; τ q T pt
τ q T ptq
(2.20)
and depicted in Figure 2.11. Substituting (2.19) into (2.20) yields:
∆1 T pt; τ q τ
1
δf0 f0
N ¸ δfi1
i!f0
pt
τ q i ti
i 2
1 ∆1 ψ ptq, 2πf0
(2.21)
where ∆1 ψ ptq is the first increment of ψ ptq. The first increment provides insight
into how the apparent time process evolves over time and, from (2.21),it can be 0 seen that for a fixed observation interval, τ , it contains a constant term τ 1 δf f0 .
This constant term can be removed by differencing successive first increments. The difference between two first increments is the second increment, denoted ∆2 T pt; τ q and is defined as:
30
Section 2.4: Receiver Non-Idealities and Associated Losses
1 Reference
Oscillator
Apparent Time (s)
0.75
∆1T (0.5; τ ) 0.5
∆1T (0.25; τ ) 0.25
τ 0 0
0.25
τ 0.5 True Time (s)
0.75
1
Figure 2.11: Example of time process of unstable oscillator indicating the first increment of the time process for τ 0.25.
31
Chapter 2: Introduction to Signals and Receivers
∆2 T pt; τ q ∆1 T pt
τ ; τ q ∆1 T pt; τ q
(2.22)
Substituting (2.21) into (2.22) yields:
∆2 T pt; τ q
δf1 2 τ f0
N ¸ δfi1
i!f0
pt
2τ qi 2 pt
τ qi
ti
i 3
1 ∆2 ψ ptq, 2πf0
(2.23)
where ∆2 ψ ptq is the second increment of ψ ptq. The second increment of the time
process is now independent of the frequency bias, δf0 , and includes a constant term, δf1 2 f0 τ ,
dependent on the frequency drift. It can easily be inferred that higher order
increments will be insensitive to the higher order deterministic oscillator drift terms. For example, the third increment of the oscillator time process, denoted ∆3 T pt; τ q,
is equal to:
∆3 T pt; τ q
δf2 3 τ f0 N ¸ δfi1
i!f0
3pt
τ q i 3 pt
2τ qi
pt
3τ qi ti
i 4
1 ∆3 ψ ptq, 2πf0
(2.24)
which is independent of both frequency offset and frequency drift. For typical oscillators the second and third increment offer sufficient insensitivity to the deterministic
oscillator instabilities, as high order drift terms (i ¥ 2) are small. Thus, the following approximation is, generally reasonable in practice:
∆i T pt; τ q
∆i ψ ptq 2πf0
for i ¥ 2
(2.25)
The variance of these increments offers a measure of the instability, ψ ptq. ∆2 T pt; τ q
specifies the variance of the difference in length of two successive time intervals, rel-
ative to two true intervals of length τ , a direct indication of the instability of the average frequency over these intervals. ∆3 T pt; τ q specifies the variance of the difference in length of two second increments and, thus, offers an indication of the
instability of the average frequency drift over these intervals. The variance of the second and third increments can be related to two standard metrics which are commonly used to assess oscillator stability. The two-sample Allan variance, denoted 2 pτ q, and the Hadamard variance, denoted σ 2 pτ q, are defined as [73, 6, 79, 8, 2, 1]: σA H
32
Section 2.4: Receiver Non-Idealities and Associated Losses
−16
10
−17
10
−18
10
Variance
σ2A(τ)
h2 = 2.0e-26 h1 = 2.5e-23 h0 = 3.9e-22 h−1 = 2.4e-21 h−2 = 2.4e-22
σ2H(τ)
−19
10
−20
10
−21
10
−3
10
−2
−1
10
0
10 10 Averaging Time τ (s)
1
10
2
10
Figure 2.12: Example Allan and Hadamard Variance traces for a typical TCXO (the fourth oscillator of Table 6.1).
2 1 2 p t; τ q E ∆ T 2τ 2 2 pτ q 6τ1 2 E ∆3T pt; τ q 2 , σH 2 pτ q σA
where the coefficients
1 2
and
1 6
(2.26) (2.27)
2 pτ q and are included by convention such that σA
2 pτ q coincide with the standard variance of ∆1 T pt; τ q in the special case where σH
p q is a white process(5) . Both metrics offer a measure of time interval stability
ψ t 2πf0
which are sufficiently independent of deterministic trends. An example of the Allan and Hadamard variances of a typical oscillator are shown in Figure 2.12. The expectations in (2.26) and (2.27), of course, cannot be ascertained exactly and must be estimated by observing the second and third increments of the time process over a sufficiently long period of time, see, for example, [6, 2] for details of this estimation procedure. These time domain measurements are obscure and yield little insight into the
characteristics of ψ ptq. A frequency domain representation of ψ ptq is, perhaps, more (5)
Note that the definition of Allan variance given in [79] differs from other definitions, for example [1]. The standard definition normalises the second increment by τ and, subsequently, scales its variance by a factor of 21 , whereas [79] normalises the second increment by 2τ and applies no scaling 2 to the variance estimate. The result is that the estimates of σA pτ q differ by a factor of two.
33
Chapter 2: Introduction to Signals and Receivers
−5
10
h2 = 2.0e-26 h1 = 2.5e-23 h0 = 3.9e-22 h−1 = 2.4e-21 h−2 = 2.4e-22
h2 f4 −10
ST (f)
10
h1 f3
−15
10
h0 f2 h−1 f
−20
10
−4
10
−2
10
h−2
0
2
10 f (Hz)
10
4
10
Figure 2.13: Example of the PSD model of oscillator phase noise of a typical TCXO (the fourth oscillator of Table 6.1). The solid curve represents the composite PSD and the line segments represent the contribution of each of the spectral elements, hα f pα 2q .
useful. An example of the PSD of T ptq is shown in Figure 2.13 which depicts that of
a typical oscillator (TCXO). For modelling purposes, an approximate spectral model of the oscillator instability is employed which typically takes the form [6, 73, 79]:
ST pf q
$ 2 ¸ ' ' &
α2 ' ' %0
hα f pα2q
for fl
¤ f fh
,
(2.28)
otherwise
where each coefficient hα denotes the intensity of each spectral component. These spectral components are also annotated in Figure 2.13. Note that the exponent of f is increased by 2 relative to the index of the hα coefficient. This is simply because the original use of this form of spectral model was based on the spectrum of the rate
of change of T ptq, more commonly known as the fractional frequency deviation of
an oscillator. This is discussed in more detail in Chapter 6. Whilst this model is rooted in the frequency domain, it can be readily related to the Allan or Hadamard domains, wherein the intensity coefficients are estimated. A thorough analysis of the interconnection between Allan and Hadamard domains and the frequency domain can be found in, for example, [6, 73, 8]. 34
Section 2.4: Receiver Non-Idealities and Associated Losses Simulation of Oscillator Phase Noise Traditionally, analysis of stochastic systems involved a trade-off between accuracy and convenience. That is, a model was chosen which faithfully reflected the phenomenon under study while remaining mathematically tractable. Until recently, only two methods of analysis were available, that of pure theory and that of traditional real-world experimentation. With the advent of fast and accessible computational resources a third approach was recognised, that of simulation. Real systems could readily be usefully modelled, generally beyond the point where a theoretical or mathematical analysis was possible, and an analysis performed through MonteCarlo simulation. Central to the Monte-Carlo method, of course, is the generation of random numbers. Although digital computers are incapable of generating truly random numbers, for the purposes of Monte-Carlo simulation, a sequence of numbers which bear some key properties of a truly random sequence will, generally, suffice. These numbers are often referred to as pseudo-random numbers and are typically generated via some recursive algorithm, for example linear feedback-shift registers and congruential generators, [43, 99]. Such generators produce uniformly distributed independent random variables. For the purposes of GNSS simulation, however, the Gaussian or Normal distribution is of interest. Many algorithms exist which can be used to produce variates of a particular distribution given variates from a uniform distribution [98, 43, 99]. In particular, the Box-Muller transform is an efficient and accurate method of producing variates from a Gaussian distribution. Unfortunately, the generation of pseudo-random, uncorrelated normally distributed random variables is insufficient. As has been shown previously, an accurate model of the receiver’s oscillator requires correlated Gaussian sequences. When considering correlated random variables it is often convenient to discuss their spectral properties. Specifically, it can readily be shown that a random sequence with PSD given by f 2 SX pf q can be generated by simply integrating samples from a random
sequence with PSD(6) given SX pf q. In the case of computer-based simulation, this
integration can be approximated by a suitably scaled summation. A random-walk random process, with PSD
1 , f2
can be generated from a white random process (with a
uniform PSD) by simply accumulating its variates. Likewise, a random process with PSD
1 f3
can be generated from a random process with PSD
1 f.
Thus, given a gen-
erator for white Gaussian random variables and one for Gaussian random variables with PSD
1 f,
a composite oscillator model of the form (2.28) can be implemented.
The generation of random variables with PSD
1 f,
however, is not an easy task.
Such processes are known as flicker noise processes and are commonly observed in natural phenomena, for example, hydrology [76] and timing [62]. For the same rea(6)
While this is true for stationary processes, the existence, or otherwise of the PSD of a nonstationary process is debatable [75], in fact, the relationship between Allan variance and the PSD of a particular random process is not absolutely defined [45].
35
Chapter 2: Introduction to Signals and Receivers sons that its correlation function is not defined, it is not possible to find a simple impulse response which will produce a sequence with a
1 f
PSD [62]. As a result,
approximations to this spectrum must be used and is the topic of ongoing research. Depending on the application, different flicker noise generators are used, some providing a satisfactory PSD over a wide range of frequencies, at the cost of a high computational burden [46, 62, 32, 128], and others which enable the fast generation of sequences, albeit with less accurate PSD [76, 7, 116]. In the case of the oscillator, assuming this composite spectral model, the spectral components with PSD of the form
1 f
and
1 f3
are ‘nested’ between spectral components with PSD
1 , 1 f0 f2
and
1 , f4
as shown in Figure 2.13. As the latter components can be generated almost exactly, the frequency range over which the
1 f
and
1 f3
processes must be accurately repre-
sented is, in fact, quite narrow. Resorting to simple, fast generators, such as [32], therefore, is reasonable. For example, a thorough description of one particularly efficient model can be found in [124].
2.5
Baseband Receiver Processing
The arithmetic operations involved in the generation of the correlator values, Im and Qm , described earlier, are executed at a nominal rate of Fs . This is the same rate at which the digitised IF signal is sampled and, so, these operations are often termed IF processing operations. Having computed Im and Qm , the remainder of the
receiver’s processing occurs at a rate of 1{TI , the rate at which these correlator values
are generated. These operations are often termed baseband processing operations. Essentially, the IF processing operations produce the correlator values from which estimates of the various received signal parameters can be made and the baseband processing operations encompass this estimation process and the refining of these estimates. The most fundamental of the baseband processing operations is that of signal acquisition. Signal acquisition is a hybrid process which entails (i) detection of the presence of a particular satellite signal in the correlator values and, if the signal is present, (ii) estimation of the signal code-phase and carrier frequency [87, 61, 108]. Typical implementations of the signal acquisition process involve the examination of the envelope or square magnitude of the complex correlator values, Im
jQm , over
a suitable range of carrier frequency and code phase values. The two-dimensional range of carrier frequency and code phase values represents what is known as the search space. If the receiver has no a priori knowledge of the received code phase then the entire code length must be searched. Similarly, without any estimate of the carrier frequency, all possible Doppler frequency values, combined with any potential oscillator frequency offset, must be searched. Although the carrier frequency and code phase are continuous variables, it is generally not practical that the acquisition 36
Section 2.5: Baseband Receiver Processing process perform a continuous search over these variables. Instead, the acquisition need provide only a coarse estimate of these parameters. Having coarsely identified the signal code phase and carrier frequency, a DLL and FLL can be employed to refine these estimates. The receiver, therefore, decimates the search space. Decimation in frequency is achieved by searching a finite number of frequency values which are evenly spaced along the frequency axis. Similarly, a finite number of code phase values are searched. The magnitude of the respective code phase and carrier frequency spacings are design choices and are often based on the coherent integration period, losses associated with residual Doppler and residual code phase and the total acquisition time [87], to name but a few. An example of the carrier frequency, code phase search space and the envelope,
a
2 Im
Q2m , for a signal in the
presence of noise is shown in Figure 2.14 (a). An example of the noise-free response of the envelope to this signal is shown in Figure 2.14 (b). Ideally, the acquisition algorithm will detect the correct peak of the envelope and provide the receiver with a valid estimate of the carrier frequency and code phase. This estimate will be accurate to within half of the magnitude of the corresponding frequency or phase decimation. The purpose of the receiver tracking algorithms is two-fold, firstly they refine these estimates, removing the residual errors and, secondly, they track the signal parameters as they change with time. In fact, the receiver tracking algorithms, generally, will provide estimates of, not just the carrier frequency and code phase, but also of the carrier phase and, depending on the receiver design, the Doppler drift rate. Tracking algorithms operate in a fundamentally different manner to the acquisition algorithm. While the acquisition algorithm examines a number of independent trials, either collectively or individually, to estimate a signal parameter, the tracking algorithms are recursive, each successive estimate being a direct function of the previous. A tracking algorithm will utilize an initial, coarse, estimate of a signal parameter to produce the correlator values, Im and Qm (or a suitable set of similar correlator values). Using these values, the residual error is estimated and filtered. This filtered estimate is then used to generate the next set of correlator values. The process of tracking the carrier parameters and of tracking the code parameters can be implemented either as two independent loops or as one composite loop. The specific details of the code tracking loops, however, are not the focus of this thesis and, so, only the problem of carrier tracking is considered here. A comprehensive tutorial style analysis of code tracking can be found in, for example [61, 114, 115, 20]. In terms of carrier tracking, the process of refining the carrier frequency estimate is typically initiated by closing an FLL. 37
Chapter 2: Introduction to Signals and Receivers
fc = 2120 Hz τ = 507.8 chips
1 p 2 + Q2 Im m
0.8 0.6 0.4 0.2
1000 5000 500
0 0
τ (chips) (a) C {N0
−5000
fc (Hz)
48 dB Hz in a full search space
1 p 2 + Q2 Im m
0.8 0.6 0.4 0.2
508 507
2500 506
τ (chips)
2000 505
1500
fc (Hz)
(b) Ideal, noise-free signal in a reduced search space. Figure 2.14: Example Acquisition Search Space.
38
Section 2.5: Baseband Receiver Processing
Correlator
r[k]
. . Current Correlator Values
c[k]
From Code Tracking Loop
OSC
.....
Local Carrier Replicas
Frequency Discriminator e
Loop Filter
NCO Figure 2.15: Generic FLL Block Diagram
2.5.1
The Frequency Lock Loop
As discussed previously, parameter tracking in a GNSS receiver is a recursive process. To track (or, at least, to attempt to track) the received signal carrier frequency, the FLL will utilize the current set of correlator values. By applying a suitable transformation or mapping to these values, in the form of the carrier frequency discriminator, the FLL will produce an estimate of the difference between the signal carrier frequency and the local replica carrier frequency. This error is then applied to the input of a filter. The function of the filter is two-fold: firstly, as the received signal and, thus, the frequency estimate, is corrupted by thermal noise, the filter is required to provide a degree of noise rejection. Secondly, appropriate design of the loop filter enables estimation of both the instantaneous frequency and, by observing sustained errors, higher order frequency dynamics. Finally, this filtered frequency error estimate is applied to the receiver NCO, to adjust the local carrier frequency for the generation of the next set of correlator values. A simplified block diagram of the FLL is presented in Figure 2.15. Frequency lock loops for GNSS applications can be categorised according to the method by which they estimate frequency error. Frequency estimates can be derived from one of two principles: that of differential power measurement and that of differential phase measurement. The differential power measurement technique exploits the relationship between the magnitude of the complex correlator value Im
jQm and the frequency error. Specifically, the relationship is embodied in the
sinc pq terms in (2.11) and (2.12). Two complex correlator values are generated, one
using a slightly high local carrier frequency and one with a slightly low local carrier frequency. The difference in the magnitude of the two complex values is then used to estimate the frequency error. This approach, as applied to GNSS receivers, is presented and analysed in, for example [60]. The differential phase measurement technique bases its frequency estimate on the difference in phase of two complex correlator values. These two complex values are separated in time, usually one correlation period, and the resulting frequency 39
Chapter 2: Introduction to Signals and Receivers estimate is simply the ratio of the difference in phase to the intervening period of time. This approach requires the generation of only one complex correlator value per correlation period, as compared to the differential power measurement technique, which requires two. To the best of the author’s knowledge, a thorough comparison of the relative performance of the two algorithms is lacking(7) , yet the differential phase measurement technique is, by far, the more frequently used approach in GNSS receivers [61, 80, 114, 121] and, therefore, is the only approach considered in this thesis. The performance of the FLL can be assessed based upon its dynamic performance: its pull-in range and its ability to track carrier frequency dynamics and also upon its noise performance: its ability to accurately track the signal carrier frequency and reject thermal noise. The pull-in range of the FLL is a function of the linear region of the frequency discriminator, which, in turn, is a function of the coherent integration period [114, 61] and must accommodate the residual frequency error of the acquisition stage. In terms of consumer applications, other than the initial transient error induced by the residual error in the acquisition estimate, thermal noise is the primary error source of the FLL, especially as the receiver may be operating in harsh signal environments. A range of closed form approximations to the carrier frequency tracking error variance of the FLL exist, corresponding to a range of different carrier frequency discriminators and under a selection of different assumptions. Primarily, these approximations assume that the discrete update FLL is well approximated by an equivalent continuous update system. A number of popular expressions are detailed in (2.29). The notation in each of the expressions has been modified from its original from such that: Bω represents the two-sided equivalent rectangular bandwidth of the loop filter, TI represents the coherent observation period and N represents the number of successive discriminator outputs that are summed prior to updating the loop filter (such that the loop update period is TL
(7)
N TI ).
Interestingly, the two different approaches actually track different signal parameters. The differential power measurement technique tracks the mean signal frequency over the correlation period. In contrast, the differential phase measurement technique tracks the difference between the mean phase over two successive correlation periods. While, in the limit as the correlation period tends toward zero, these two tracked parameters converge to the instantaneous signal frequency, they are, nonetheless, different. Moreover, in terms of optimality, while both techniques are sub-optimal, the differential power technique, perhaps, bears a greater resemblance to the maximum-likelihood frequency estimator [79] than the differential phase technique.
40
Section 2.5: Baseband Receiver Processing 8
10
(a) (b) (c) (d) (e) (f)
7
10
6
σ2ω (rad2/s2)
10
5
10
4
10
3
10
2
10
1
10
20
25
30
35 C/N0 (dB Hz)
40
45
Figure 2.16: Comparison of FLL noise performance estimates for Bω
σω2
$ ' ' ' ' ' ' ' ' ' ' &
' ' ' ' ' ' ' ' ' ' %
(iq
(iiq
(iiiq
(iv q
(v q
2F Bω C N0
TI2
p { q
1
p q TI2
p { q
{ q
1
Bω C N0 1 N Bω TI
p { qp
Bω TI2 C N0
p { q
1 N
Bω 1 2Pe 2 TI2 C N0
p
p
q
p { q
p { q
N 1 2TI C N0
q
(2.29)
p { q
p { q
1 N
1 2TI C N0
1 N
1 2TI C N0
4 Hz and TI 0.001 s.
1 TI C N0
Bω N N 1 2 TI2 C N0
50
1 2TI C N0
p { q
1
p p q
8TI Bω Pe 1 1.5Pe 1 2Pe 2
q
The origins of all of these expressions are not entirely clear, some are given without proof, while others are attributed to documents describing various marginally similar systems. The first expression, (2.29) piq, can be found in [61, 121] and describes the performance of the FLL, regardless of the discriminator. The factor F
is intended to account for the variation in behaviour between high and low C {N0 ,
assuming a value of 1.0 for high C {N0 values and 2.0 for low C {N0 . The exact threshold at which this value should change, however, is not specified. An example of this variance estimate is depicted in Figures 2.16 (a) and (b) for F
1.0
and 2.0, respectively. The performance of the cross-product discriminator is given by (2.29) piiq, a derivation of which can be found in [80]. An example of this estimate is depicted in Figure 2.16 (c). A slightly different expression, describing,
again, the cross-product discriminator, is (2.29) piiiq, originally presented in [85]
and depicted in Figure 2.16 (d). An adaptation of this estimate, found in [114], 41
Chapter 2: Introduction to Signals and Receivers again pertaining to the cross-product discriminator, is given by (2.29) piv q and de-
picted in Figure 2.16 (e). Finally, (2.29) pv q, first presented in [86], describes the
performance of the decision-directed cross-product discriminator, and is depicted in
Figure 2.16 (f). The term Pe denotes the probability of error of the bit-sign decision, and is given by: Pe
eT
LC
{N0
(2.30)
It is clear from Figure 2.16 that these five expression differ significantly, even those which attempt to describe the same FLL configuration. The approaches,
in particular that of (2.29) piiq and pv q, begin with quite accurate models and
intermediate results, yet all fail to accurately model two phenomena. Firstly, they
all apply the assumption that the variance of the frequency can be directly related to the equivalent noise bandwidth. This assumption implies that the noise incident on the loop filter is white, an assumption which, as shown in Chapter 4, does not
always apply. In particular, for low C {N0 conditions, when the non-linearities of the discriminators become pronounced, the spectrum of the corrupting noise can be distorted. This effect, it will be shown, must be considered when predicting the FLL noise performance. Secondly, the correlation between successive discriminator estimates is neglected. While this correlation is simply ignored in [80], it is claimed that it is equal to zero in [86]. It will be shown in Chapter 4 that this correlation is non-zero and, in fact, as large as -0.5, in some cases. The correlation of the discriminator, it will be shown, has a large impact on the resultant FLL performance and is central to
understanding the varying FLL performance with changing C {N0 conditions. Also,
in the case of (2.29) pv q, this correlation has an impact on the related probability
of error, Pe . While (2.30) accurately reflects the probability of error of a signal corrupted by white noise, it does not account for a correlated noise. The impact of this inaccuracy is discussed in Chapter 4, wherein a modified, more accurate, expression is presented.
With regard to (2.29) piq, it will be shown in Chapter 4 that the assumption that
all discriminators can be approximated by the same function is invalid. In fact, the performance of each discriminator exhibits a unique relationship with the prevailing C {N0 . This feature, it will be shown, implies that the choice of discriminator can be optimised, based on the operating conditions. Moreover, it is, perhaps, n¨aive to assume that the variation in performance of the FLL, across a wide range of C {N0 conditions, can be embodied in a simple binary decision between F
2.5.2
1 and F 2.
The Phase Lock Loop
Similar to the FLL, the PLL is a recursive estimator. Applying a suitable discriminator function to a selection of correlator values (Im and Qm ), the PLL can estimate 42
Section 2.5: Baseband Receiver Processing
Im
r[k]
Correlator
Phase Discriminator
Qm
e
c[k]
From Code Tracking Loop
cos
OSC
sin
Loop Filter
NCO Figure 2.17: Generic PLL Block Diagram
the current carrier phase error. This carrier phase error is then filtered and applied to the NCO to produce the next set of correlator values. As the carrier phase is influenced by all of the higher order carrier effects (frequency, frequency drift etc.) it can be used to produce estimates of both the phase and of these higher order effects. A suitably designed PLL loop filter, therefore, can be useful in the estimation of PVT as it can, potentially, provide good quality estimates of the relative satelliteto-user velocity and acceleration. The general structure of the PLL is depicted in Figure 2.17. The function of the PLL is not dissimilar to that of the FLL: it must adequately track the signal dynamics, providing a recursively estimated control frequency (e) to the NCO, while providing sufficient thermal noise rejection, that the PLL remains synchronised with the received signal. The increased sensitivity to the carrier parameters that the PLL offers, while useful in the estimate of PVT, comes at a cost. For example, the degree of carrier phase dynamics required to induce loss of lock in a PLL is, generally, lower than that required to induced loss of lock in an FLL. Receiver effects, such as oscillator-induced phase perturbations, which can generally be neglected in FLL analysis, must be considered in the design of the PLL. The first operation performed by the PLL is that of phase error estimation. The choice of phase estimator is not a simple one, and many optimal and suboptimal estimators have been proposed, see for example [74, 94, 61, 114, 70]. The ultimate choice is, generally, not the globally optimal estimator, but, rather, is an estimator that effects a suitable trade-off between implementation constraints, algorithmic efficiency and estimation performance. As the discriminator choice will be based on the target application, factors such as the prevailing carrier-to-noise ratio, predicted phase dynamics and the presence or absence of data modulation often play a part [61, 114, 42, 79]. Despite the vast range of possible discriminator functions, typically, in the case of GNSS receives, the choice is reduced to a small set of popular discriminators [61, 114]. As will be shown in Chapter 5, the discriminator is one of the primary design features which dictate the overall PLL performance and, 43
Chapter 2: Introduction to Signals and Receivers while detailed analyses of its effect on the tracking capabilities of the continuous update PLL are plentiful [47, 127, 101, 72, 78, 109], such treatments, specific to the GNSS architecture, considering long coherent integration periods and low carrierto-noise ratios, are scarce. Performance analysis of the PLL falls into two main categories, dynamic performance and noise performance (of course, in some cases, a hybrid performance analysis is necessary). Typically, in terms of consumer grade GNSS receivers, only a limited dynamic performance is required as the peak accelerations of a civilian user are generally small [65, 23, 69], this will be discussed further in Chapter 6. In such cases, a linearised analysis of the PLL dynamic response can be conducted, to ascertain the expected peak transient errors. For example, [74] presents a comprehensive tutorial on dynamic models and transient responses of digital PLLs. Receivers which may need to tolerate high carrier phase dynamics, for example, aviation or military applications, are not considered here as, typically, the receiver architecture will differ significantly from that of a consumer grade receiver. In addition to deterministic phase dynamics, the PLL must accommodate the stochastic phase perturbations induced by the instability of the local oscillator. These phase anomalies, discussed in Section 2.4.3, can pose a significant threat to the PLL, particularly under weak-signal conditions. For example, when the received carrier-to-noise ratio is low, the bandwidth of the PLL is often reduced to alleviate thermal-noise-induced tracking errors. This reduced bandwidth, however, effects a reduced dynamic tracking capability which can cause oscillator induced tracking errors to become comparable to, or exceed, those induced by thermal noise. Using the standard power-spectral density model described in Section 2.4.3, the variance of this oscillator induced tracking error can be readily estimated [15, 66, 54, 79, 61]. In terms of noise performance, a linear analysis of the PLL is trivial, and the related tracking error variance can readily be estimated [61, 114, 79, 42, 29]. In the case of weak signals, however, a simple linear model is not always readily available. For example, it is well documented that the performance of some common discriminator functions degrades with reduced C {N0 (for example, they exhibit reduced gain
and a contracted linear region [61, 114, 101]), yet a concise characterisation of this
degradation, for many PLL configurations, is lacking. This issue will be examined in Chapter 5. One example of hybrid performance analysis, mentioned previously, is that of the PLL acquisition stage. The behaviour of the PLL as it initially acquires the carrier phase and frequency of the received signal, subsequent to handover from the FLL, includes deterministic trends, but is also heavily influenced by the presence of corrupting thermal noise. A number of approaches to the mathematical modelling of this process have been taken, utilizing a variety of simplified models or specific PLL configurations. These approaches typically employ stochastic calculus methods which aim to identify the time-varying distribution of the carrier phase error as 44
Section 2.5: Baseband Receiver Processing the PLL undergoes this acquisition transient. Such treatments include, for example, [72, 79, 78, 127]. While they are accurate, these approaches can be unwieldy and may offer little insight into the influence of various design parameters on the ultimate PLL performance. Given the abundance of computation resources currently available to designers, numerical methods are proving a popular alternative [127, 123]. The latter approach is utilized in Chapter 6 to perform such a hybrid performance analysis of the acquisition time of the PLL. The second major PLL design consideration is that of the loop filter. The loop filter performs a number of key tasks (comparable to those of the FLL). It must strive to reject thermal noise while emitting the useful error signal. Furthermore, it must establish running estimates of sustained carrier phase dynamics (Doppler, oscillator frequency bias and Doppler drift etc.). Traditionally, when PLLs were predominantly continuous-time systems, the loop filter design was based on Laplace-domain PLL models. These models offer a simple relationship between key loop parameters (bandwidth and damping) and the filter implementation (the filter coefficients). Not wishing to part with a wealth of accumulated design theory, these continuous-time design practices were maintained and regularly applied to discrete-update systems [61, 121, 114, 80]. In the case of weak-signal GNSS receivers, where long (relative to the loop bandwidth) coherent integration periods are commonly used, the weakness of the continuous-update approximation of discrete update loops became clear. This sparked renewed interest in loop filter design which produced a host of discrete-update specific design routines, for example [104] tabulates a selection of design rules for discrete update PLL design. While these tabulated results are useful, the decimation is coarse. Rather than resort to interpolation of the results in [104], Chapter 4 offers alternative, approximate functions for the generation of filter coefficients for certain discrete update systems. In reality, the determination of the filter coefficients which satisfy a particular PLL response (bandwidth and damping), is, perhaps, the easy part. The act of choosing the bandwidth and a damping parameter may, in fact, be more challenging. Fortunately, for the designer (although, perhaps unfortunate for the user), the choice must strive to satisfy two conflicting requirements: the need for high bandwidth and a rapid dynamic response, to satisfy the relative dynamics of the satellite and the user; and the need for bandwidth minimisation for the purposes of noise rejection. These conflicting requirements are often encapsulated in a simple rule of thumb [61]: 3σP LL
a
3
σn2
σv2
σo2
θd
¤ θT hr ,
(2.31)
where σP2 LL represents the total PLL carrier phase error. The quantities σn2 , σv2 and σo2 respectively denote the tracking error variance induced by: thermal noise; vibration induced oscillator phase perturbations; and stochastic oscillator instability. The transient error resulting from deterministic phase dynamics is represented by 45
Chapter 2: Introduction to Signals and Receivers
23 dB Hz
0
10
28 dB Hz 38 dB Hz
33 dB Hz
43 dB Hz σ2δθ (rad2)
−1
10
48 dB Hz
−2
10
h0 = 3.9e-22 h−1 = 2.4e-21 h−2 = 2.4e-22
−3
10
0
10
1
2
10
10 Bθ (Hz)
Figure 2.18: A selection of thermal noise and assuming line intersects
simple, single-parameter optimisations of the PLL design considering and oscillator phase noise, for a range of carrier-to-noise-floor ratios a second order, standard-underdamped loop filter. The broken black 2 each σδθ curve at its respective local minima.
θd . The phase error threshold, denoted θT hr , is a theoretical upper limit on the total carrier phase error, beyond which the PLL is liable to lose lock. For the
coherent and non-coherent discriminators, these are often assumed to be π {2 and π {4, respectively. While this type of expression effects a reasonable cost function,
by which the effectiveness of various design can be measured, it offers little insight into the preferred design. Nonetheless, it is often applied, in one form or another, in numerical design optimisations, for example [80, 54], wherein a trade-off between the the oscillator and the thermal noise constraints is considered. An equivalent, single parameter optimisation (across design bandwidth), is depicted in Figure 2.18. Of course, one need not always resort to a numerical method of finding the optimum filter design. The problem of filter design can be solved analytically, by simply minimising the cost function (the tracking error variance) with respect to some design parameters. For example, [79] presents an optimisation of the second order, continuous update PLL, with respect to thermal noise and flicker phase noise (h1 ) and [15] presents an optimisation of the Kalman filter with respect to white
frequency modulation phase noise (h0 ) and thermal noise. Approaches such as these will achieve optimality within the constraints imposed in the problem formulation, the second order PLL and the Kalman filter, respectively. In some cases, further performance improvements can be achieved if no filter form is stipulated. One 46
Section 2.6: Conclusions such approach is that of Wiener filter design [22], and will be discussed further in Chapter 6 in terms of optimal filter design for the PLL.
2.6
Conclusions
This chapter has introduced the GNSS signals which are of primary interest in this thesis, namely, the GPS L1 C/A, the GIOVE-A E1 B/C and GIOVE-B E1 B/C signals. The respective modulation schemes and spreading code chip materialisation and correlation properties have been studied. Given the intended scope of this work, the propagation channel of the typical civilian user has been examined. It is clear that the signals incident on a typical consumer grade receiver will potentially be subject to severe attenuation and distortion, with received carrier-to-noise-floor ratios ranging from 48 dB Hz to as little as 15 dB Hz. With respect to performance of the receiver components, it has been shown that the consumer receiver will exhibit signal losses associated with the front end filter, the ADC and the local frequency reference. These losses serve to compound the problem of low received signal-to-noise ratio. To maximise the performance of the receiver, given these limited resources, the receiver operation must be fully characterised and the associated losses, and their implications, fully understood. In terms of the IF receiver processing, this analysis of the receiver quantisation and filtering losses is presented in Chapter 3. In terms of the baseband processes, the FLL and the PLL when subject to weak received power levels, exhibit nonlinear behaviour. Central to this non-linearity, in both cases, is the discriminator. Enhanced modelling accuracy can be achieved through characterising this non-linearity, with a view to developing more comprehensive performance models. Chapters 4 and 5 examine the carrier phase and carrier frequency discriminators and investigate their role in the performance of the tracking loop. Chapter 6 will then utilize the PLL model in the design of the PLL for steady state operation. Finally, before proceeding with the main contributions of this thesis, it is worth emphasising the limitations of the signal and receiver models discussed here. To form a useful system model some simplifying assumptions must be made, some of which are, perhaps, reasonable and others, perhaps, crude. Indeed, one can always contrive a situation in which a particular assumption becomes invalid. Nonetheless, for the purposes of this work, the following assumptions were made: There is only one single satellite signal incident on the antenna: this assump-
tion implies that the effects of inter-system and intra-system interference can be neglected. This is deemed reasonable as it is assumed that the cross correlation properties of the spreading codes offer sufficient isolation between the received signals. There is only a direct LOS signal path: the effects of multi-path propagation
47
Chapter 2: Introduction to Signals and Receivers on the cross correlation of the received code and the local code are neglected. It is assumed that the effects of fading can be embodied in a single loss coefficient applied to the received power. There is no RF interference: this assumption is similar to the first assumption;
not only is there no interference caused by other GNSS signals, there is neither wideband, narrowband, continuous or pulsed interference present in the received signal ensemble. The dynamics of the user are minimal and commensurate with a typical pedes-
trian or civilian user. This implies that the received carrier frequency is within 10 kHz of its nominal frequency and that the carrier frequency drift is less than 2 Hz/s in magnitude. This assumption is discussed further in Chapter 6. The effects of the ionosphere and of the troposphere are neglected. It is
assumed that the phase and amplitude anomalies induced by propagation through the ionosphere and troposphere are negligible compared to other propagation effects and receiver non-idealities. It is assumed that the GNSS receiver employs an ideal anti-alias filter at IF
followed by real IF sampling (as opposed to complex, or quadrature, sampling). Also, it is assumed that the receiver local replica signals can be represented with absolute precision. In terms of baseband receiver processing, the type of sampling (real or complex) is irrelevant. In terms of IF signal processing it can be shown that the two approaches are equivalent, in terms of signal representation (although correlation of complex sampled signals can incur an additional computational overhead). These simplifying assumptions apply to all of the work presented in the following chapters. The justification for, and validity of, each assumption will be discussed as the implications of each particular assumption arise. Following the route, along which the received signal propagates through the receiver, this work will next examine, in Chapter 3, the receiver intermediate frequency operations and the generation of the baseband receiver parameters and proceeding, in Chapters 4, 5 and 6, to consider a selection of baseband receiver processes.
48
Chapter 3
Quantisation and Filtering Losses As outlined in the previous chapter, position, velocity and time estimates in GNSS receivers are all formed from GNSS signal parameter estimates. These signal parameters are estimated using some manifestation of a matched filter. In the case of modern GNSS receivers, which are invariably digital receivers (being either dedicated digital hardware receivers or, more recently, hybrid hardware/software receivers [40, 3, 18]) the matched filter operation is implemented in its digital form: the DMF. Chapters 4, 5 and 6 will discuss receiver functions which operate on these signal estimates (i.e. carrier phase / frequency tracking), while this chapter examines, in detail, the formation of these signal parameter estimates. Typically, consumer grade receivers use both a low resolution quantiser (e.g. one or two bits) and a low sample rate, necessitating a narrow front-end filter, which results in reduced signal quality. The impact of this loss of signal quality on the performance of a receiver’s DMF is examined in this chapter. A mathematical model for the receiver’s front-end and the signal down-conversion process is presented in Section 3.1. This model is utilised in Section 3.2, wherein the ideal receiver is considered and the concept of processing losses is defined with reference to the ideal receiver. Section 3.3 presents a thorough analysis of the front-end filter induced processing loss for a range of typical receiver configurations. A general treatment of B-bit symmetric quantisation is provided in Section 3.4, including novel results pertaining to the propagation of AWGN through a DMF in the presence of both front-end filtering and quantisation. In Section 3.5, these novel results are applied to the problem of front-end filter bandwidth and center-frequency optimisation for GNSS receivers, specifically for one-, two- and three-bit quantisation and the reception of the GPS L1 C/A, GIOVE-A B/C and GIOVE-B B/C signals. Finally, in Section 3.6, both a simulation-based and real signal-based validation of these results is presented, including a comparison of these new loss estimation techniques with classical loss models. 49
Chapter 3: Quantisation and Filtering Losses
rRF (t ) Quantiser Filter
Filter
Filter
r[k ]
AMP
LNA
FS OSC
A
QBg [x]
Freq. Synth
Figure 3.1: Block Diagram of Front-End.
3.1
Signal and System Model
The system model used in this analysis assumes that a single satellite signal, sRF ptq,
distorted by AWGN, n ptq, is incident on the receiver’s antenna. As discussed in
Chapter 2, a receiver will, typically, perform initial amplification and frequency selection at RF and, subsequently, mix the signal to IF, in one or multiple stages, in the analogue domain [40]. The signal incident on the antenna can be expressed as: rRF ptq sRF ptq
sRF ptq
?
n ptq
2P cos pωRF t
θ q c p t τ q d pt τ q ,
(3.1)
where P is the received signal power and ωRF and θ represent the RF carrier frequency and phase, respectively. Functions c pt τ q and d pt τ q respectively model
the signal spreading code sequence and data sequence, delayed by time τ . The noise term, n ptq, is assumed to be white and to have a two-sided PSD of N0 {2 W/Hz. The process of down conversion to IF, filtering, sampling and quantisation cannot be clearly partitioned into individual steps. The filtering process begins with the frequency selectivity of the antenna. After the first amplification stage, at the LNA, the received signal is typically filtered, mixed to an IF and again filtered to remove the mixer-induced double frequency terms. A further amplification stage may then be applied to the signal, usually of higher gain than the LNA. This entire process may be iterated until the desired IF is achieved. The process of sampling and quantisation is one single step: a continuous signal is sampled and immediately rendered to a discrete amplitude. A block diagram of an example of this process is depicted in Figure 3.1. Since an exact mathematical representation of this process would be quite cumbersome, it is useful to avail of a more compact, mathematically equivalent, model. In this simplified model the down-conversion process is conducted, without filtering, directly from RF to the final IF. This signal is then passed through an ideal anti-alias filter and sampled to produce a discrete-time, continuous-amplitude signal. The filtering operations are represented by one discrete-time filter, applied to the sampled, discrete-time, signal. Finally, the filtered, discrete-time signal is quantised to a discrete amplitude. Figure 3.2 depicts the simplified down-conversion process. The result of ideal frequency selection and down-conversion to IF is a continuous 50
Section 3.1: Signal and System Model
rRF (t ) Amplifier, Mixer and Anti-alias Filter
H f ( e jω )
rIF (t )
rIF [k ] FS
rf [k ] IF Filter
Quantiser
r[k ] A
QBg [x]
Figure 3.2: Block Diagram of Equivalent Front-End Model.
signal, rIF ptq, centered at ωc . The center frequency, ωc , is equal to the nominal IF
center frequency, ωIF , plus a residual Doppler frequency. This signal is sampled at a rate of FS Hz (where the sample period is denoted Ts ), producing the discrete
time sequence, rIF rk s. A discrete time filter is next applied to this signal to pro-
duce rf rk s. Finally, this filtered sequence is quantised, producing the signal r rk s.
Without loss of generality, the gain of the down-conversion and filtering stages can be assumed to be unity, and so rIF rk s is given by: rIF rk s sIF rk s
sIF rk s
?
n rk s
2P cos pωc kTs
θq c pkTs τ q d pkTs τ q .
(3.2)
The equivalent filter in Figure 3.2 represents the combined effects of the antenna and multiple filtering stages of the front-end on the amplitude and phase of rIF rk s. It
has a z-domain transfer function denoted by Hf pz q, spectrum Hf pejω q and impulse
response hf rk s. Generally, Hf pejω q is a bandpass filter, centered at ωIF . Given
hf rk s, the filtered signal can be expressed as(1) : rf rk s prIF
hf q rks .
(3.3)
The bandwidth of Hf pejω q depends greatly on the quality of the receiver and on
the intended application. Consumer grade GPS receivers, for example, intended for the GPS L1 C/A signal, will have a bandwidth of, typically, 2 MHz, corresponding to a main lobe width of approximately 2 MHz(2) . For the GIOVE signals, GIOVE-A The notation pa bq rks represents the convolution of a and b. Typically, consumer-grade, or mobile handset embedded, GNSS receivers use a single-chip solution. Such single-chip receivers incorporate the front-end signal processing, multiple (twenty or more) acquisition and tracking channels and provide the user with a navigation solution. Current examples of these chipsets are: the SiRF SiRFstarIV GSD4t [102], the Ublox UBX-G6010 GPS and GALILEO single chip receiver [113] and the Broadcom BCM4750 - Single-Chip AGPS Solution [21]. While these chipsets provide the majority of the receiver functionality, they rely on some external components such as the antenna and RF matching circuitry, a power supply, one or more external oscillators (generally a TCXO) and an RF filter. This filter is, generally, implemented as a SAW filter. Ublox provides a comprehensive application note for their receivers [112], wherein a number recommendations regarding the SAW filter are made. Providers of some such filters include muRata [82] and EPCOS [37], see, for example, the EPCOS B9444 [38] or the muRata SAFEA1G57KD0F00 [83], both of which claim to provide a usable bandwidth of 2 MHZ centred at 1575.42 MHZ. (1) (2)
51
Chapter 3: Quantisation and Filtering Losses
1 N
r[k ]
∑
y[m]
2 exp{− j(ωˆ c kTS + θˆ )} c(kTS − τˆ) Figure 3.3: Block Diagram of Correlator.
B/C and GIOVE-B B/C, the bandwidth will generally be of the order of 4 MHz, corresponding to the two main lobes of the signals. For applications which require high degrees of accuracy, and those which employ advanced code tracking techniques [19], a wider front-end bandwidth is often employed. The quantisation operation is represented here as a memoryless, piecewise con-
tinuous function, denoted QBg r s, such that: A
r rk s QBg rrf rk ss , A
(3.4)
where a B bit, symmetric quantiser is assumed and the scalar Ag represents the gain of the quantiser AGC, the details of the quantiser will be discussed in detail in Section 3.4. As discussed in Chapter 2, a DMF is typically applied to this signal to extract the parameters of interest, generally one or more of the carrier frequency, carrier phase, code phase and data sign. This filter (or, equivalently, correlator), shown in Figure 3.3, operates on N successive samples of r rk s, throughout the duration of the correlation period, TI , where N Ts
TI .
This approximate relationship exists
because the sample period, Ts , is often not a divisor of the code period(3) . When Ts is not an integer fraction of TI , a receiver will, typically, dither the value of N
TI holds. The value of N , however, is sufficiently large that, for the purposes of this analysis, the approximation N Ts TI such that, on average, the relationship N Ts
is reasonable. To avail of the autocorrelation properties of the spreading code, the coherent integration period, TI , is chosen to be an integer multiple of the spreading code period. The output of this correlator can be expressed as: y rms
N 1 ? 1 ¸ c pkTs τˆq 2 ej pωˆ c kTs N k0
q rrk
θˆ
mN s.
(3.5)
Estimates of ωc , θ and τ are, respectively, represented by ω ˆ c , θˆ and τˆ. These estimates are, generally, adapted by the receiver so as to maximise |y rk s|, the magnitude of the correlator output y rk s. The details of the formation of these estimates will be
(3) In fact, there are a number of reasons for intentionally making this the case, for example see [18].
52
Section 3.1: Signal and System Model discussed further in Chapters 4 and 5. Note here that although the effect of Doppler on the carrier has been accounted for (in the variables ωc and ω ˆ c ) no corresponding term exists for the effect on the code. Indeed, Doppler does induce an apparent code rate which is not equal to the true code rate and most receivers will estimate and track this apparent code rate(4) . The effect of this code rate variation is, however, negligible in terms of received signal-to-noise ratio [61] for relatively short coherent integration periods (for example, less than 20 ms), and will, for notational convenience, be neglected in this chapter. The filter described in (3.5) can be expressed as the convolution of r rk s and the
impulse response hc rk s. Assuming that the correlator output is given by:
y rms pr hc q rN s
¸
r rk s hc rN
ks,
(3.6)
k
then, by analogy with (3.5), it can be shown that: hc rk s
w rk s c ppN N
kq Ts τˆq
?
2ej pωˆ c pN kqTs
q,
θˆ
(3.7)
where: w rk s
$ &1
for 0 ¤ k
%0
otherwise
N
(3.8)
is a window function which only emits samples from the preceding TI seconds. In the above expressions, and those which follow, it is assumed that, where no limits are stated in a summation, the summation is to be evaluated from
8 to 8.
The
variable y rms is complex valued, the real and imaginary parts of which can be
represented by the real valued variables Im and Qm , respectively. Combining (3.6)
and (3.7), y rms can be expressed in terms of the received signal, the equivalent
front-end filter, the quantiser and the correlator impulse response:
y rms pr hc q rN s
QAB rrf s hc rN s QAB rrIF hf s hc rN s . g
g
(3.9)
(4) Interestingly, a comparison between the estimates of the code rate and the carrier rate (ˆ ωc ) can be used to detect false carrier frequency lock [61].
53
Chapter 3: Quantisation and Filtering Losses
3.2
The Ideal DMF and The Loss Coefficient
It is useful to consider a special case of the correlation operations defined above in which the receiver components are ideal. In such a case the front-end filter is an “all-pass” filter and the quantiser has infinite resolution and, so, is simply a unit gain; that is:
1 hf rk s δn A QB rxs x.
Hf ejω
(3.10)
g
Inserting these assumptions into (3.9), and noting that δk is the identity convolution element, yields: y rms prIF
hcq rN s .
(3.11)
The mean value of y rms is equal to the noise-free value and is given by: E ry rmss µy
pE rrIF s hcq rN s psIF hcq rN s .
(3.12)
Substituting (3.2) and (3.7) into (3.12), neglecting the double frequency terms
(since they will be filtered out by hc rk s) and simplifying the summation, we find:
µy
?
ˆ c ωc q N Ts sin 12 pω ˆ Rcode pτˆ τ q ej ppθθq P 1 ˆ c ωc q Ts N sin 2 pω
1 2
pωˆc ωc qpN 1qTs q ,
(3.13)
where Rcode pτ q is the autocorrelation function of the spreading code and is given by:
Rcode pτ q E rc ptq c pt τ qs .
(3.14)
The differences between the true signal parameters and the receiver’s estimates of these parameters can be expressed as:
ωˆc ωc δθ θˆ θ δτ τˆ τ,
δωc
and, furthermore, we note that: 54
Section 3.2: The Ideal DMF and The Loss Coefficient
sin N
δωc N Ts 2 δωc Ts sin 2
sinc
δωc N Ts 2
pN 1qTs N Ts TI ,
and so (3.13) reduces to:
µy
?
P sinc
δωc TI 2
Rcode pδτ q e
j δθ
δωc TI 2
(3.15)
.
The sincp•q term represents an attenuation due to frequency error, typically due
to residual Doppler and oscillator frequency noise and is equal to unity for perfect
frequency alignment. The Rcode p•q term represents the correlation gain brought about by the correlation across one or more full code periods and has a maximum
value of unity (for zero code error). The complex exponential term represents the average phase difference between the incoming signal carrier and the locally generated carrier, containing both the initial carrier phase error, δθ, and the mean phase error over the correlation period, induced by the initial frequency error, δωc . The variance of y rms is given by: Var ry rmss σy2
E py rms µy q py rms µy q
(3.16)
.
From (3.11), noting that, in the ideal case, the system is linear, the noise component of y rms is given by:
y rms E ry rmss pn hc q rN s .
(3.17)
Substituting (3.17) into (3.16) and simplifying yields:
σy2
E ppn hcq rN sq pn hcq rN s E
¸
¸¸
n rk s hc rN
¸
ks
k
n rls hc rN
ls
l
hc rN k s h rN ls E rn rk s n rlss c
k
l
k
l
¸¸
hc rN
ks hc rN ls Rn rk ls ,
where Rn rk s is the autocorrelation function of the noise. Letting k proceeding to make some substitution of variables: 55
(3.18)
Ñk
l, and
Chapter 3: Quantisation and Filtering Losses
σy2
¸
Rn rk s
k
¸
hc rN
k ls hc rN ls
l
¸
Rn rk s Rc rk s ,
(3.19)
k
where Rc rk s denotes the autocorrelation function of the correlator impulse response and is given by:
Rc rk s
¸ l
hc rls hc rl
N22
¸
ls ej ωˆ c kTs c plTs τˆq c ppl k q Ts τˆq
w rls w rk
l
N 1k 2ej ωˆ c kTs ¸
N2
For the special case of k simplifies to:
ks
c plTs τˆq c ppl k q Ts τˆq .
(3.20)
l 0
0, which proves useful in the evaluation of σy2, Rc rks
Rc r0s
N 1 2 ¸ pc plTs τˆqq2 N 2 l0
N2 Rcode p0q
N2 .
(3.21)
As discussed previously, the noise sequence, n rk s, is white with PSD equal to
N0 2
W/Hz and so its autocorrelation function can readily be shown to be given by:
Rn rk s
N0 δk . 2Ts
(3.22)
Substituting (3.21) and (3.22) into (3.19), the variance of y rk s for the ideal receiver, with an “all-pass” front-end filter and infinite quantiser resolution, is given by: 56
Section 3.2: The Ideal DMF and The Loss Coefficient
σy2
¸
Rn rk s Rc rk s
k
¸
N0 2T
s k
δk Rc rk s
N0 2 2T N s
N0 . TI
(3.23)
It is useful to adopt the ideal receiver as a benchmark, against which actual receiver implementations and configurations can be compared. One option is to consider the non-coherent signal to noise ratio of the correlator output. This metric
is defined here as the ratio of the square magnitude of the mean of y rk s to its variance, and is given by:
SNRnc
µy µy . σy2
(3.24)
SNRnc is non-coherent in the sense that it is independent of the phase of y rk s, that ˆ For the ideal receiver, the maximum achievable value of is, it is independent of θ. SNRnc , w.r.t. the receiver variables ω ˆ c and τˆ, denoted here SNRIdeal nc , is given by:
SNRIdeal nc
max τˆ,ˆ ω c
µy µy σy2
P sinc
2
δωc TI 2
PNTI .
2 Rcode
pδτ q NTI 0 δωc 0,δτ 0
(3.25)
0
Another useful metric is the coherent signal to noise ratio of the correlator output. This metric is defined as the ratio of the square of the mean value of the real part of y rk s to the variance of its real part:
E r< ty rk susq pVar r< ty rksus . 2
SNRc
(3.26)
To find an expression for this metric, the properties of the real part of y rk s must
initially be examined. From (3.12) it can be seen that the expectation of the real part of y rk s is simply the real part of its mean and so, for the ideal case, using (3.15), yields:
57
Chapter 3: Quantisation and Filtering Losses
< t µy u µ i
?
"
*
δωc TI < P sinc δω2cTI Rcode pδτ q ej δθ 2
? δωc TI δωc TI . P sinc 2 Rcode pδτ q cos δθ 2
(3.27)
By following a similar approach to (3.18) and (3.19), the variance of the real part of y rk s can be found:
Var r< ty rk sus E
21
p< ty rksu µiq2
¸
Rn rk s Rc rk s
k
21 σy2.
(3.28)
For the ideal receiver the maximum achievable value of SNRc , w.r.t. the receiver variables ω ˆ c , θˆ and τˆ, denoted here SNRIdeal , is given by: c
SNRIdeal c
?
max ˆ
τˆ,θ,ˆ ωc
P sinc
µ2i 1 2 2 σy
δωc TI 2
Rcode pδτ q cos δθ
δωc TI 2
2PNTI .
2
2TI N0 δωc 0,δθ0,δτ 0 (3.29)
0
This value is equal to the signal-to-noise ratio of the real part of y rk s when the
received signal and the local replica are perfectly synchronised and is equal to twice the non-coherent signal-to-noise ratio.
Given these signal-to-noise ratio metrics for the ideal receiver, the loss imposed by the non-idealities of any real receiver can be quantified by comparison with the ideal case. A loss coefficient can be defined as: SNRnc c SNRIdeal Ideal SNRnc SNRc N0 SNRnc P T I N0 SNRc 2P T . I
L
(3.30)
This loss coefficient is a positive real number and represents the apparent reduction 58
Section 3.3: Front-End Filtering Effects in received signal-to-noise ratio that has been induced by a particular receiver, rela-
tive to the ideal receiver (i.e. for the ideal receiver L 1). This metric will be used to quantify the degradation in signal quality caused by filtering and quantisation in the following sections.
3.3
Front-End Filtering Effects
The front-end filter affects y rms in two ways: firstly, it distorts and attenuates the
mean value of y rms and, secondly, it introduces correlation into the noise component of y rms. These effects, and their implications for the value of SNRc and L, are
examined here in the absence of quantisation (i.e. QBg rxs x). It will be shown in A
Chapters 4, 5 and 6 that the metric SNRc is particularly useful in the prediction of closed loop carrier tracking performance.
The propagation of the signal mean through the equivalent front-end filter can
be examined using (3.9), whereby an expression for y rms in the presence of filtering but, in the absence of quantisation, is given by:
y rms pr hc q rN s
prf hcq rN s prIF hf hcq rN s .
3.3.1
(3.31)
Propagation of the Mean
Following a similar procedure to Section 3.2, and employing (3.31), µy is given by:
E ry rmss psIF
hf hcq rN s ,
(3.32)
where:
psIF hf hcq rls
¸
hf rns
n
¸
hc rk ns sIF rl k s.
k
This particular expression is cumbersome but can be simplified by noting that the window function in hc reduces the summation over k to the interval n ¤ k
such that:
¸
hc rk ns sIF rl k s
k
?
¤ pN 1
P sinc
δωc TI 2
ej pδθ
δωc TI 2
nTs ωc
59
q Rcode rpl nqTs δτ s .
(3.33)
nqq,
Chapter 3: Quantisation and Filtering Losses Fs Fc Tc Bf
25.0 3.123 0.978 2.0
MHz MHz µs MHz
Table 3.1: Signal and Receiver Parameters
Substituting (3.33) into (3.32) yields:
µy
psIF hf hcq rN s
? δωc TI P sinc 2 ejpδθ
δωc TI 2
q
N¸1
hf rns ejnTs ωc Rcode rpN
nqTs δτ s.
n 0
(3.34)
The significance of (3.34) lies in fact that the summation on the right hand side represents the convolution of the spreading code autocorrelation function, the signal carrier frequency and the impulse response of the front-end filter. This term embodies the impact of hf rk s on µy .
To further explore the effects of the hf rk s on µy , it is useful to consider an
example. An eighth order, bandpass, elliptic filter with a 0.1 dB pass-band ripple
and a 20 dB stop-band attenuation is assumed at the receiver’s front-end. The specific details of this example are presented in Table 3.1 in which Bf denotes the bandwidth of the filter. Assuming that the receiver can exactly replicate the signal
carrier frequency (i.e. δωc
0) and that the correlator value can be perfectly
normalised by the received signal power, then, for this example, we can assume that: µy
ejδθ
N¸1
hf rns ejnTs ωc Rcode rpN
nqTs δτ s.
(3.35)
n 0
Figure 3.4 shows both the real and imaginary parts of µy and its magnitude (or
0 for this example signal and receiver. Also shown for the ideal case, in which hf rk s δk . Three filtering ef-
envelope) versus δτ , with δθ is the envelope of µy
fects are immediately apparent. Firstly, the shape of the cross-correlation function
has changed, the function has become ‘rounded’ and is not symmetrical about its peak value. Secondly, the magnitude of the mean has reduced with respect to the ideal case. This is due to both the removal of the received signal power which lies outside the passband of the front-end filter and the distortion of the received code by the front-end filter. Thirdly, the values of δτ at which the magnitude of µy is maximised and the argument of µy at this point have changed. The front-end filter has introduced a delay in the signal path and also a rotation of the carrier phase. It is important to note here that the square magnitude of µy is always less than 60
Section 3.3: Front-End Filtering Effects
1
Ideal |µy| ℜ{µy}
0.8
ℑ{µy} 0.6
|µy|
0.4 0.2 0 −2
−1
0
1 δτ
2
3
4
T−1 (chips) c
Figure 3.4: Normalised in-phase and quadrature correlator mean values and envelope for the GPS L1 C/A signal with receiver configuration described in Table 3.1.
or, at best, equal to the total signal power which passes through the front-end filter. The correlator is no longer, strictly speaking, a matched filter, and so the local code replica will not always correlate exactly with the received code. To implement a
truly matched filter, a filter identical to Hf pω q would have to be applied to the local code replica, prior to correlation with the received signal. This implies the following inequality:
max |µy |
2
δτ,δωc
¤
P 2π
»π
π
ˆ c q|2 Scode pω q dω, |Hf pω ω
(3.36)
where Scode pω q F tRcode pτ qu is the PSD of c ptq given by the Fourier transform of Rcode pτ q.
The cause of the inequality in (3.36) is the relative phase between different
spectral components of c ptq in the filtered received code and the unfiltered local
replica code. The local code replica will have no phase distortion and, so, when the receiver maximizes the magnitude of µy , it aligns the ensemble of the spectral components of the filtered and unfiltered codes by adjusting the time delay, τˆ. Of course, phase and delay are linearly related by frequency, and so when the phase distortion induced by Hf pω q is nonlinear, some spectral components may not align
exactly. Equality holds when the phase response of Hf pω q is linear.
Another, rather interesting, effect of the front-end filter is that it introduces a coupling between received carrier phase error, δθ, and received code phase error, δτ . This, perhaps unintuitive, effect is evident in Figure 3.4 where it can be seen that the ratio of the real to the imaginary part of µy is not constant with changing δτ . This 61
Chapter 3: Quantisation and Filtering Losses
3
∠ µy (rad)
2 1 0 −1 −2 −3 −2
−1
0
1 δτ
2
3
4
T−1 (chips) c
Figure 3.5: The angle of the correlator mean value for the GPS L1 C/A signal with receiver configuration described in Table 3.1.
is also evident by considering the summation in (3.35), wherein the coefficients of the complex exponential in the summation are samples of the code autocorrelation function, which is a function of δτ . This introduces a coupling between carrier phase tracking and code phase tracking. Changing the receiver variable τˆ will not
only induce an attenuation of the magnitude of µy (via Rcode pδτ q), but it will also cause it to rotate, inadvertently changing the apparent carrier phase error. While
this result is an interesting anomaly, in practice it is relatively unimportant. For the reception of GNSS signals, to maximise use of the transmitted signal power, the code phase error must be kept relatively small (for example,
12
Tc ). The relationship
between the code phase error and the angle of µy is, thus, of interest only within this region. As can be seen from Figure 3.5, the angle of µy is relatively constant with
¡ Tc, the angle of µy does change dramatically with changing δτ , however, in this region Rcode pδτ q is so small as to changing δτ in the range
Tc ¤ δτ ¤ Tc.
For δτ
make the signal useless. As it is relatively constant, when processing the received ˆ GNSS signal, this phase angle can be absorbed into the receiver phase estimate, θ, as it simply represents the apparent phase of the received signal, including both the true signal phase and the filter-induced phase. It is reasonable, therefore, to neglect this code-carrier coupling effect. To see how µy varies with Bf , the magnitude plot of Figure 3.4 has been repeated for a variety of different Bf in Figure 3.6. In this figure, Fs has been increased to 55 MHz and Fc has been increased to 6 MHz to accommodate the full range of Bf . It is evident that the peak magnitude decreases monotonically with reduced front-end bandwidth. An example of how the peak magnitude varies with Bf is shown in Figure 3.7. The absolute value of the slope in the region of the peak value also 62
Section 3.3: Front-End Filtering Effects
1
Ideal 10 MHz 5 MHz 4 MHz 2 MHz
0.8
|µy|
0.6 0.4 0.2 0 −1.5
−1
−0.5
0
0.5 δτ
1
1.5
2
2.5
T−1 (chips) c
Figure 3.6: |µy | versus δτ Tc1 for a selection of front-end filter bandwidths.
reduces with reducing Bf . More detail on these results and on their implications for SNRnc , SNRc and L will be given in Section 3.5.
3.3.2
Propagation of the Variance
As before, the noise component of y rms is given by y rms µy and so the variance
of y rms is given by:
σy2
E py rms µy q py rms µy q E pn hf hc rN sq pn hf hc rN sq .
(3.37)
Expanding and simplifying yields:
σy2
E
¸¸ l
¸¸
ks
p
k
hc rps hf rq ps n rN
q
qs
¸¸ hc rls h rps hf rq ps hf rk ls E rn rN q s n rN k ss c
l
hc rls hf rk ls n rN
¸¸
p
¸¸ l
Letting k
q
hc rls hc rps
p
Ñk
k
¸¸ q
hf rq ps hf rk ls Rn rk q s .
k
q and noting that Rn rk s is delta correlated: 63
(3.38)
Chapter 3: Quantisation and Filtering Losses
1 0.95
|µy|
0.9 0.85 0.8 0.75 0
2
4
6
8
10 12 Bf (M Hz)
14
16
18
20
Figure 3.7: max |µy | versus Bf . δτ,δωc
σy2
¸¸
hc rls hc rps
l
p
Rn r0s Rn r0s
¸¸
¸¸ q
hf rq ps hf rk
q ls Rn rk s
k
¸ hc rls h rps hf rq ps hf rq ls
c
l
p
l
p
¸¸
q
hc rls hc rps Rf rp ls ,
(3.39)
where Rf rk s is the autocorrelation function of the filter impulse response, hf rk s. Letting l Ñ l
p:
σy2
Rn r0s
¸
Rn r0s
¸
l
Rf rls
¸
hc rl
ps hc rps
p
Rf rls Rc rls .
(3.40)
l
Equation (3.40) implies that the variance of y rk s is proportional to the thermal
noise floor at the antenna and is shaped by both the front-end filter and the correlator. This has some interesting implications and will now be examined. Substituting the correlator impulse response into (3.40) yields: 64
Section 3.3: Front-End Filtering Effects
σy2
RnNr0s
¸
Rf rls Rcode rls ej ωˆ c lTs
l
Rn r0s ¸ ˆ c lTs q , Rf rls Rcode rls cos pω N l
where, as sin pxq is an odd function, and the summation is from term reduces to its (even) cos pxq component.
(3.41)
8 to 8, the ex
Generally, for a CDMA system, the code autocorrelation function will be approximately zero outside of a one chip region around a zero code phase offset(5) and the summation over l can be reduced to the region(6)
rFsTcs.
This significantly
reduces the number of operations required to evaluate (3.41) also, noting that Rf rk s and Rcode rk s are even functions, we find:
σy2
Rn r0s Rf r0s N
¸
rFs Tc s
2
ˆ c lTs q . Rf rls Rcode rls cos pω
(3.42)
l 1
Examining either (3.41) or (3.8), it is clear that Rcode rk s has a very significant
impact on σy2 . In the case that Rf rk s is not delta correlated, the more rapidly Rcode rk s decays to zero, the less of a contribution the summation in (3.41) will
make. This effect is depicted in Figure 3.8 in the frequency domain, wherein the incident noise at IF is depicted in frame (a). Filtering this noise, using hf rk s,
results in a band-pass noise, centered at ωc , depicted in frame (b). The local carrier replica is used to mix this signal to baseband, resulting in the low-pass noise PSD of frame (c). Multiplication of this signal by the local code replica has the effect of spreading this noise, as shown in frame (d). Recall that, unlike the signal component
of rIF rk s, the noise component has not been previously modulated by a spreading
code. Multiplication by c rkTs τˆs spreads the noise and displaces noise power from the vicinity of 0 Hz and thus, upon application of the moving average filter (ΣN k1 ), reduces the variance of y rk s. As the output of the moving average filter is sub-
sampled(7) by a factor of N , all of the power in the narrow-band low-pass noise process of frame (e), is aliased into the band
t 2T1
I
u. This y rms has variance
1 2TI
equal to the variance of the narrow-band low-pass noise process of frame (e), but (5) CDMA codes for spread spectrum, such as the GPS L1 C/A gold codes, or the ‘memory’ codes of the GIOVE-A E1 B/C and GIOVE-B E1 B/C, do have a non-zero autocorrelation outside of the region of the main peak. These so called side-lobes are generally 30 dB lower than the main peak and so do not contribute significantly to the summation in (3.41). (6) rxs denotes the ceiling of x equal to the nearest integer that is not less than x. (7) By sub-sampling, it is implied that the output of the MA filter (the correlator) is sampled at a rate which is less than the rate at which it is generated, and without the use of an anti-alias filter. Thus, Nyquist criteria are not upheld and aliasing occurs. The entire power of the original process is aliased back into the new passband, dictated by the new sample rate. As the process is sub-sampled by quite a large factor (greater than 1000), the effect of aliasing renders the new process, y rns, approximately white.
65
Chapter 3: Quantisation and Filtering Losses
N0 2
h f [k]
ωc
0 (a) White noise incident on front-end filter
(b) Filtered, band-pass noise centered at ωc ˆ
e − j(ωˆkTs +θ ) c[kTs − τˆ]
0 (d) Spread-spectrum noise
0 (c) Low-pass noise
N
∑
TI
k =1
y[m] sub-sample
0 (e) Narrow-band, low-pass noise Figure 3.8: PSD of the noise component of y rms throughout correlation process.
has a white PSD. As noted above, the front-end filter has a significant impact on σy2 as it dictates the autocorrelation function of the noise. Examining, again, (3.41), it is evident that the slower the decay of the filter impulse response (i.e. the narrower the bandwidth) the lower the value of σ y 2 . The exact relationship between Bf and σ y 2 is not
linear, however, and is depicted in Figure 3.9, wherein σ y 2 , normalised by that of the ideal case, is plotted against Bf . The normalised variance approaches unity
as Bf increases to its maximum value (dictated by the Nyquist criterion) and falls sharply with decreasing Bf , for Bf values compable to the main lobe width of the spreading code PSD.
3.3.3
Filtering Losses
In terms of filtering losses, the effects of the front-end filter on µy and on σy2 suggest conflicting design choices for the receiver’s front-end filter. From Section 3.3.1, it has been shown that µy is maximised by maximising Bf , which increases SNRc . In contrast, in Section 3.3.2, it has been shown that increasing Bf increases σy2 , which reduces SNRc . Jointly analysing both effects yields insight into the true filtering losses. An exmple plot of the filtering loss coefficient, as per (3.30), is 66
Section 3.4: Quantisation Effects
σ2y TI / N0
0.95
0.9
0.85
0.8 0
2
4
6
8
10 12 Bf (M Hz)
14
16
18
20
Figure 3.9: Normalised correlator variance versus Bf .
shown in Figure 3.10. It can be seen that, unlike |µy | (Figure 3.7), the loss coefficient
does not increase monotonically with increasing Bf and there exist local maximima, roughly correspoding to the nulls between each side lobe of the code PSD. The loss coefficient reduces dramatically with reducing Bf for Bf values less than the main width of the main lobe of the spreading code PSD (approximately 2 MHz, in this case) and is roughly constant with changing Bf for Bf
¡ 8 MHz. For this example
receiver, it is clear that, for the purposes of SNRc maximisation, a front-end filter bandwidth of approximately 2 MHz is sufficient, with 4 MHz or 5 MHz, perhaps, being a reasonable compromise. Increasing Bf beyond this value does not yield a discernable SNRc improvement. In terms of the shape of the correlation peak, in the context of code tracking, however, it may be beneficial to increase Bf to achieve a ’sharper’ correlation peak. This aspect, however, is beyond the scope of this analysis and is not persued here. It must be stressed that the results of Figures 3.7, 3.9 and 3.10 are particular to the sample rate and IF center frequency chosen in this example and, while they are representative of a typical GNSS receiver, results will vary as these parameters are varied. The impact of Fc and Fs variation on the loss coefficient will be examined in detail in Section 3.5.
3.4
Quantisation Effects
The effects of quantisation on the mean, variance and SNRc (or SNRnc ) of y rms are
considered in this section. The specific model assumed is that of a signal which is
quantised to 2B evenly spaced levels centred around 0 volts. The model also assumes that, prior to quantisation, the signal is scaled by a gain, Ag , which represents 67
Chapter 3: Quantisation and Filtering Losses
0 −0.2
L (dB)
−0.4 −0.6 −0.8 −1 −1.2 −1.4 0
2
4
6
8
10 12 Bf (MHz)
14
16
18
20
Figure 3.10: Loss coefficient (L) versus Bf .
the AGC of the front end. Such a configuration, known as a B-bit symmetric quantiser, is depicted in Figure 3.11. This quantisation operation can be represented mathematically by: A QBg
rxs
2
B
1
L ¸
2 i
where L
L
u pAg x i q ,
2pB1q 1 and u rxs denotes the Heaviside step function.
(3.43) For consumer
applications, one- or two-bit quantisers are most commonly employed while higher resolution quantisers (using more bits) are generally used only for specialised applications which require high precision and robustness. Although the theoretical framework presented here will consider a general B-bit quantiser, explicit numerical results will be presented only for one- to three-bit quantisers.
3.4.1
Propagation of the Mean
The theory of the propagation of the signal mean through a B-bit symmetric quantiser has been well documented in [16] and only a brief overview of the details will be given here. As an introductory example, the propagation of the signal mean through a 1-bit quantiser will be considered. In this case, the quantiser can be expressed as: Q1 g rxs 2u pAg xq 1. A
(3.44)
Such a quantiser is known as a ’hard-limiter’ as it resembles a very large gain which saturates at a finite limiting amplitude, in this case 1. Incidentally, as will be shown
later, this quantiser is independent of the AGC gain. From (3.4), the quantised sample when a 1-bit quantiser is used, is given by: 68
Section 3.4: Quantisation Effects
A
QBg [x]
5
3
1 1 -2
-1 -1
2
Ag x
-3
-5 Figure 3.11: Transfer characteristic of the B-bit symmetric quantiser with AGC gain Ag .
r rk s 2u pAg rf rk sq 1.
(3.45)
As r rk s has only two possible values in this case (i.e. +1 and -1), the mean of r rk s can be evaluated by examining the two probabilities:
P pr rk s 1q P pAg rf rk s ¥ 0q
P pr rk s 1q P pAg rf rk s 0q .
(3.46)
Recall that rf rns is the sum of a deterministic component (the instantaneous
mean), psIF
hf q rks,
and a zero mean additive Gaussian noise component with
autocorrelation function
denoted:
N0 2Ts Rf
rks.
For convenience, these two components will be
psIF hf q rks N0 σf2 Rf r0s . 2T
µf
(3.47) (3.48)
s
The probabilities of (3.46) can now be well approximated through the use of the 69
Chapter 3: Quantisation and Filtering Losses random variable: r rk s
N pµf , σf q, which yields the well known results(8) [94, 16]:
P pr rk s 1q
1 2
P pr rk s 1q
1 1 2
erf
1 erf
?µf 2σf
?µf 2σf
(3.49)
,
and so the mean value of r rns is: E rr rnss 1 P pr rk s 1q
12
erf
1 erf
?µf 2σf
p1q P pr rks 1q
?µf 2σf
1 2
1 erf
?µf 2σf
(3.50)
.
While (3.50) is exact, evaluation of erf p•q can be rather inconvenient, and so it is
useful to consider a linearisation of this expression, about the operating point. Since sIF rk s has zero mean, the linearisation is about the origin:
E rr rnss µf
c
B
E r n µf µ f 0
B
r r ss
2 µf . π σf
(3.51)
Note that, in this case, as the only threshold is at zero volts, the presence of the AGC does not influence the value of the mean. The accuracy of this linearisation is dependent on the ratio
µf σf .
Numerically it can readily be shown that the approxi-
mation is accurate to within 2% for
µf σf
¤ 0.35 and to within 10% for µσ ¤ 0.8. f
f
To
examine the suitability of this approximation, the GPS L1 C/A signal is considered, taking the worst case scenario (the highest value of
µf σf )
of a high signal strength (50
dB Hz) and a narrow front-end bandwidth (2.0 MHz). Referring to Table 3.1 and
assuming a received C {N0 of 50 dB Hz, it can readily be shown that
µf σf
0.3106,
implying that the approximation is valid to within 2%, which is reasonable. With
increased front-end bandwidth and reduced C {N0 , the approximation becomes more accurate.
In the case of multi-bit quantisation a similar approach can be used to evaluate the mean value of r rk s. The analysis becomes more complicated and the value of (8)
Where erf pxq is the Gauss error function and is defined as: erf pxq
70
x ³ ?2π et2 dt. 0
Section 3.4: Quantisation Effects Ag has a significant influence on the performance of the quantiser. A thorough derivation of the mean of a B-bit symmetric quantiser is given in [16] and repeated below for convenience. This expression is based on the premise that
µf σf
is small, µ
as above, and that integration under a Gaussian p.d.f across short intervals (2 σff ) can be approximated by a summation. In the context of typical GNSS reception,
as discussed above, these approximations are reasonable and result in the following expression for the quantiser mean: E rr rk ss
?2µf 2πσf
1
L ¸
#
i2 exp 2 2 2 2σf Ag i1
+
.
(3.52)
It is interesting to note that the mean term, µf , can be factored out of (3.52), such that the net effect of the quantiser on the received signal mean is that of a scalar gain. This gain, denoted here by KQ , is a function of the number of bits, B, used for the quantisation, the AGC gain, Ag , and the variance of the noise incident on the quantiser (σf2 ): E rr rk ss KQ µf
ñ KQ
?2 2πσf
1
L ¸
#
i2 2 exp 2 2 2σf Ag i1
+
.
(3.53)
A plot of KQ versus AG for the receiver of Table 3.1 and, again, assuming a C {N0
of 50 dB Hz, for the one-, two- and three-bit quantiser in Figure 3.12. In light of this, an estimate of µy in the presence of both filtering and quantisation is given by:
µy
pKQsIF hf hcq rN s
? δωc TI j pδθ
KQ
3.4.2
P sinc
2
e
δωc TI 2
q
N¸1
hf rns ejnTs ωc Rcode rpN
nqTs δτ s.
n 0
(3.54)
Propagation of Gaussian Sequences Through Non-linear Functions
Unlike the propagation of the signal mean through the quantiser, the propagation of the noise component of the received signal cannot be represented by a simple gain. Not only is the variance of the quantised noise altered by the non-linearity of the quantiser but the entire autocorrelation function is altered. This altered autocorrelation function, and its interaction with the correlator impulse response, has a significant impact on the performance of the DMF. To explain this effect, propagation of Gaussian sequences through non-linear functions must first be examined. 71
Chapter 3: Quantisation and Filtering Losses
0.7
1−bit 2−bit
0.6
3−bit KQ
0.5 0.4 0.3 0.2 0
5
10 Ag σf
15
20
Figure 3.12: KQ versus Ag σf for the one-, two- and three bit quantiser assuming the receiver configuration of Table 3.1 and a C {N0 of 50 dB Hz
As an example, a unit-variance, low-pass Gaussian process will be considered. This sequence was quantised using a one-, two-, and three-bit quantiser, and the PSD and autocorrelation functions of the original sequence, and each of the quantised sequences, were estimated. The PSD and autocorrelation functions of this process are shown in Figure 3.13. As the process of quantisation maps a continuous voltage to a set of binary numbers, the measure of absolute power is lost. For the purposes of comparison, the variance (power) has been normalised such that the total power in each of the four sequences is equal. For each of the three quantised sequences, the autocorrelation function decays more rapidly with increasing sample lag than the original sequence. This decorrelation corresponds to a ’whitening’ of the original, correlated sequence. As can be seen from the PSDs in Figure 3.13, power is displaced from the original pass-band, and is spread across the entire spectrum. It is also clear that the effect is more pronounced when fewer bits are used in the quantiser, or the more non-linear the quantiser function. The case of one bit quantisation, which can be compared to extreme clipping (occurring, for example, when a linear amplifier is driven into saturation) has been well documented in [119], wherein the effect on the noise PSD is referred to as ’Bottle-necking’. When the noise process is Gaussian, as is the case here, the autocorrelation function of the quantised noise can be related to the autocorrelation function of the continuous noise. The exact relationship depends on the characteristics of the quantiser. Results specific to the one-bit quantiser (which can be likened to extreme clipping in a linear amplifier, or a hard-limiter) have been developed [119, 9, 10] and 72
Section 3.4: Quantisation Effects
Continuous 1−bit 2−bit 3−bit
PSD (Normalised)
1 0.8 0.6 0.4 0.2 0 0
0.05
0.1 0.15 0.2 Normalised Frequency
0.25
0.3
(a) Power spectral density of a low-pass process
Continuous 1−bit 2−bit 3−bit
1 0.8 ρ[k]
0.6 0.4 0.2 0 −0.2 0
10
20 30 Sample Lag
40
(b) Autocorrelation function of a low-pass process Figure 3.13: PSD and autocorrelation function for a simple low-pass process with continuous amplitude and for one-, two- and three-bit quantisation.
73
50
Chapter 3: Quantisation and Filtering Losses can be applied to the problem. A more general solution, however, is preferable and is given by [93] equation (21):
B Q pτ q Bρpτ qk kR
p q
p q
8 8
x 2xr12ρρ ppττqqsx x 1 ρ2 pτ q
» 8 » 8 f pkq x f pkq x exp 1 2 2 1 a
2π
2 1
2 2
1 2
2
dx1 dx2 .
If f1 pxq and f2 pxq are two memoryless, nonlinear functions and x1
x2 k th
x pt
(3.55)
x ptq and
τ q are samples of a Gaussian random process, then (3.55) defines the
partial derivative of the cross-correlation function of f1 px1 q and f2 px2 q, denoted
RQ pτ q, with respect to the correlation coefficient of x ptq, denoted ρ pτ q, and the k th partial derivatives of f1 pxq and f2 pxq with respect to x, respectively denoted
pkq pxq and f pkq pxq.
f1
2
To apply this relationship to the problem of quantisation of correlated noise,
f1 pxq and f2 pxq must be related to the quantiser function. As (3.55) is defined in
terms of a zero mean, unit variance noise, f1 pxq and f2 pxq can be defined in terms
of (3.43) such that:
f1 pxq f2 pxq QBg rσf xs , A
(3.56)
which have first derivative given by
p1q p1q f px q f px q 1
2
L B QA rσ xs 2A σ ¸ δ pAg σf x iq . g f Bx B f iL g
(3.57)
The correlation coefficient of the incident noise (i.e. the normalised autocorrelation function of the noise prior to quantisaion) is defined as: Rf rns . Rf r0s
ρ pτ q
(3.58)
Substituting (3.57) into (3.55) and performing the double integration, the first partial derivative of the autocorrelation function of the quantised process is found to be:
pq r p qs
L L exp ¸ BRQ pτ q 4 ¸ a B ρ pτ q 2π 1 ρ2 pτ q iL kL and, subsequently, integrating with respect to ρ pτ q yields:
1 i2 k2 2ρ τ ik A2g σf2 2 1 ρ2 τ
RQ pτ q
L » ρpτ q exp ¸
L 2 ¸ π iL kL
0
A 1σ ?
2 2 g f
r s
(3.59)
i2 k2 2rik 2 1 r2
1 r2
dr
CQ ,
(3.60)
where CQ is the constant of integration. The value of CQ can be found by considering the case ρ pτ q 0 (i.e. x1 and x2 are independent) and evaluating RQ pτ q. 74
Section 3.4: Quantisation Effects
RQ pτ q E rf1 px1 q f1 px2 qs
RQ p
τ
q
p q0
ρ τ
E rf1 px1qs E rf1 px2qs E rf1 px1qs2 .
(3.61)
Noting that x1 is a unit variance Gaussian random variable and that f1 pxq is odd (i.e. f1 pxq f1 pxq), then:
E rf1 px1 qs
»8
8
f1 pxqe
x2 2
dx
0 6 CQ 0.
(3.62)
In the case of a one-bit quantiser, it has been shown that (3.60) admits a simple closed form solution [119, 10, 93]: RQ pτ q
2 sin1 ρ pτ q . π
(3.63)
Figure 3.14 shows the relationship between the correlation coefficient of noise incident on a quantiser and the normalised autocorrelation function of the quantised noise for a one-, two- or three-bit quantiser. As has been shown previously, in Figure 3.13, it can be seen, again, that the correlation of the quantised sequence decreases more rapidly with decreased correlation of the input sequence, the fewer the number of bits used. As expected, the more bits are used, the “more linear” the quantiser function becomes and the more the autocorrelation function of the quantised sequence resembles that of the non-quantised sequence.
3.4.3
Propagation of the Variance
Quantisation is a non-linear operation and so the principle of superposition does not
apply to the propagation of the mean and noise components of rf rk s through the quantiser, that is:
QBg rrf rk ss QBg rpsIF A
A
hf q rkss 75
QBg rpn hf q rk ss . A
(3.64)
Chapter 3: Quantisation and Filtering Losses
1
x=y 3−bit 2−bit
RQ(τ) / RQ(0)
0.8
1−bit
0.6
0.4
0.2
0 0
0.2
0.4
ρ(τ)
0.6
0.8
1
Figure 3.14: Normalised autocorrelation function of the quantised sequence versus non-quantised correlation coefficient for the one-, two- and three-bit quantiser.
Considering, first, the autocorrelation function of rf rk s: E rrf rk s rf rk
lss E
psIF hf q rks psIF hf q rk
ls
psIF hf q rks pn hf q rk ls psIF hf q rk ls pn hf q rks pn hf q rks pn hf q rk ls psIF hf q rks psIF hf q rk
ls
Rn rls Rf rls
(3.65)
and considering the typical signal to noise ratio of GNSS signals, it is reasonable to assume that the right hand side of (3.65) is dominated by the second term,
Rn rls Rf rls. Under this assumption, the autocorrelation function of the quantised sequence (no longer zero mean) can be approximately related, via (3.60), to the correlation coefficient of the noise component of the non-quantised sequence. Although superposition does not hold, it is still possible to define a signal component / noise component decomposition of the quantised signal plus noise sequence, similar to that of the non-quantised sequence. It is defined as: r rk s KQ µf
nQ rk s
(3.66)
where nQ rk s represents the additive noise which distorts the quantised signal. Con76
Section 3.4: Quantisation Effects sidering, now, the autocorrelation of r rk s:
2 psIF lss E KQ
E rr rk s r rk
hf q rks psIF hf q rk
KQ psIF
hf q rks nQ rks KQ psIF hf q rk ls nQ rk nQ rk s nQ rk ls
ls ls (3.67)
and noting that, by definition, nQ rk s is zero mean, and, similar to the non-quantised
signal, that the signal is dominated by the noise component, then
E rr rk s r rk
2 lss KQ psIF
RQ rls ,
hf q rks psIF hf q rk
ls
RQ rls (3.68)
where RQ rls is the autocorrelation function of nq rk s. Thus, specifically, RQ rls is
approximately related to Rf rls via (3.60) and (3.58), neglecting the mean values of the non-quantised and quantised sequences. This approximation is valid for weak signals (such as GNSS signals), distorted by additive Gaussian noise, as is the case for GNSS signals. Care must be taken, however, when applying this approximation to strong signals, as the noise component in the sequence autocorrelation functions may no longer be dominant. To examine the propagation of the thermal noise through the front-end filter,
quantiser and correlator, the variance of y rms can be found by analogy to (3.41):
σy2
N1
¸
N1
¸
RQ rls Rc rls
l
RQ rls Rcode rls ej ωˆ c lTs
l
1 ¸ ˆ c lTs q RQ rls Rcode rls cos pω N l
σy2
N1 RQ r0s
¸
rFs Tc s
2
(3.69)
ˆ c lTs q . RQ rls Rcode rls cos pω
(3.70)
l 1
Figure 3.15 shows the relationship between σy2 and Ag for one-, two-, and threebit quantisation for both the unfiltered and filtered cases. The filtered case uses the 2 MHz filter described in Table 3.1. It can be seen that the filter has a large impact on σy2 , reducing it considerably, when compared with the unfiltered case for a given Ag . As will be shown in Section 3.4.4, the correlator variance is a function of both 77
Chapter 3: Quantisation and Filtering Losses
3−bit 2−bit 1−bit
1
N σ2y
10
0
10
0
1
2
3 Ag σf
4
5
6
Figure 3.15: Correlator variance versus Ag for one-, two-, and three-bit quantisation for both the unfiltered (solid lines) and filtered (broken lines) cases.
the front-end filter bandwidth and its center frequency, but, typically, the variance reduces with reducing filter bandwidth. The reduced variance is a result of the interaction between the local code and carrier replica autocorrelation functions and the autocorrelation function of the noise, which is determined by the front-end filter and the quantiser, similar to the process described in Section 3.3.2 and Figure 3.8.
3.4.4
Joint Quantisation and Filtering Loss
Having considered the effect of quantisation on the signal mean and on the correlation function of the filtered noise, it is possible to evaluate the total signal-to-noise ratio loss incurred in the receiver front-end filter and quantiser. Again, only the one-, two- and three-bit quantisers are considered here, yet the theory presented can readily be extended to any B-bit symmetric quantiser. As has been shown in
Sections 3.4.1 and 3.4.3, the mean and variance of y rk s are dependent on Ag σf , thus, the total loss will also be considered as a function of Ag σf .
A plot of the loss coefficient, calculated using (3.30), (3.26) and (3.29), for the one-, two- and three-bit quantisers, is shown in Figure 3.16. Both the filtered (broken lines) and unfiltered (solid lines) cases are considered, using the receiver configuration of Table 3.1 for the filtered case. Figure 3.16 reveals two interesting features of the losses incurred through filtering and quantisation. Firstly, it is clear that the total loss is a function of Ag σf for the two- and three-bit cases when the signal is both filtered and unfiltered and that there exists an optimal value for Ag σf which maximises this loss coefficient. This phenomena has been well documented in [16] for the unfiltered cases, wherein optimal values for Ag σf are presented. It is evident, however, that the optimal value 78
Section 3.4: Quantisation Effects
0 1−bit 2−bit 3−bit
L (dB)
−0.5
−1
−1.5
−2 0
2
4
Ag σf
6
8
10
Figure 3.16: Quantisation and filtering loss versus Ag σf for one-, two-, and three-bit quantisation for both the unfiltered (solid lines) and filtered (broken lines) cases.
of Ag σf , in the filtered case, differs slightly from the unfiltered case, the optimal value being slightly lower in the filtered case. This difference is quite small and can be neglected without incurring a significant loss, especially considering that the loss curves are relatively smooth and do not fall sharply to the right of the peak value. The second significant feature of Figure 3.16 is the fact that the signal-to-noiseratio loss incurred through both filtering and quantisation is not always greater than the loss incurred through quantisation alone. Recall from (3.30) that a loss coefficient of unity (L
1) implies ideal receiver performance and L 0 implies total signal
loss. Thus, as L approaches 1 (or 0 dB), the better the receiver performance. For
the one-bit cases the value of L is greater for the filtered signal than it is for the unfiltered signal. A concise analysis of this effect is presented in, for example, [28]. This, perhaps unintuitive, result implies that the presence of two receiver nonidealities (finite front-end bandwidth and signal amplitude quantisation) results in superior signal recovery than the presence of signal quantisation alone. The cause of this is the decorrelation of the noise which occurs when the signal is quantised. Noise power is displaced from the narrow frequency band in which the signal lies, and spread across the entire spectrum, from
Fs{2 to Fs{2, similar to Figure 3.13.
Thus, when the front-end bandwidth is sufficiently narrow, yet not so narrow as to completely eliminate the useful signal, then, if one-bit quantisation is employed,
SNRnc and SNRc can be greater that they would be were the signal quantised but not filtered. For multi-bit quantisation (B
¡ 1) this effect no longer renders a higher
loss coefficient for the filtered case than for the unfiltered case, yet it does reduce the 79
Chapter 3: Quantisation and Filtering Losses discrepancy between the unfiltered and filtered cases for their respective optimal Ag values. In this example, for two-bit quantisation, the presence of the front-end filter has no nett effect on the total loss. A thorough analysis of filtering and quantisation loss for various front-end filters is presented in the next section.
3.5
Front-end Bandwidth and Centre Frequency Optimisation
In light of the results presented in Section 3.4.4, this section examines the impact of filter bandwidth, Bf , IF center frequency, Fc , and, where applicable, the AGC gain, Ag , on the performance of the receiver. The theory developed in Section 3.4.4 is used to evaluate the processing loss incurred by a range of ideal filters for one-, two- and three-bit quantisation.
The filter Hf ejω is assumed to be a “brick wall” (see Figure 3.17), zero-phase
filter. Although such a filter is not realisable, it provides a fair comparison over a wide range of bandwidths and centre frequencies, as variations in gain roll-off and phase non-linearity need not be considered. The frequency response of the filter is defined as follows:
jω
Hf e where FP 1
$ &1
%
0
for 2πFP 1
¤ ω ¤ 2πFP 2
otherwise
Fc Bf {2 and FP 2 Fc
(3.71)
,
Bf {2. The frequency response of such a
filter is depicted in Figure 3.17. Taking the inverse Fourier transform of (3.71), the discrete impulse response of the filter is found to be: hf rns
$ &2 F P1
p
FP 2q {Fs
% sinp2πnFP 1 {Fs qsinp2πnFP 2 {Fs q πn
for n 0 otherwise
.
(3.72)
The Nyquist criterion restricts the range of Fc and Bf pairs to a triangular
region on the tBf , Fc u plane(9) , depicted in Figure 3.18. The aim of this analysis is
to examine the surface defined by the loss coefficient over this region in order to gain an insight into the effects of filtering and quantisation on the correlation operation. Using (3.54), (3.70) and (3.30), the loss coefficient was evaluated for all allowable pairs of Fc and Bf . In the case of two- and three-bit quantisation, the value of Ag has been numerically optimised to maximise the processing loss coefficient, although it is generally found to be close to the values presented in [16]. Figure 3.19 depicts the loss surface for a one-bit quantiser and a sample rate (9)
Fc and Bf must be chosen carefully such that the pass-band is contained in the positive spectrum B Bf F2s . Fc and Bf , of course, are between 0 and F2s . This implies that Fc ¡ 2f and that Fc 2 non-negative. These inequalities define the triangular region depicted in Figure 3.18
80
Section 3.5: Front-end Filter Optimisation
Hf [f ]
Bf
1
FP1
FC
FP2
f
Figure 3.17: Frequency response of ideal IF filter, Hf ej2πf .
Fc Fs 2
{Bf ,Fc} pairs
Fs 4
0 0
Fs 2
Bf
Figure 3.18: Region of usable tBf , Fc u pairs for the ideal brick wall, front-end filter.
81
Chapter 3: Quantisation and Filtering Losses
−1.4
L (dB)
−1.6 −1.8 −2 −2.2
1
2
3
1
4
3
2
4
Fc (MHz)
Bf (MHz)
Figure 3.19: Loss Surface for One-Bit Quantisation GPS C/A (Fs = 10 MHz).
−0.6
L (dB)
−0.8 −1 −1.2 −1.4 −1.6
4
3
2
1
1
Fc (MHz)
3
2 Bf (MHz)
Figure 3.20: Loss Surface for Two-Bit Quantisation GPS C/A (Fs = 10 MHz).
82
4
Section 3.5: Front-end Filter Optimisation
−0.4
L (dB)
−0.6 −0.8 −1 −1.2 4
3
2
1
3
2
1
Fc (MHz)
Bf (MHz)
Figure 3.21: Loss Surface for Three-Bit Quantisation GPS C/A (Fs = 10 MHz).
of 10 MHz. It can be seen that as the value of Bf is increased from zero, the loss coefficient increases, up to a point, before reducing smoothly as the bandwidth increases to Fs {2. The choice of center frequency also has an impact on the loss
coefficient, L, which is found to be symmetrical about Fs {4.
This response can be envisaged as the product of two factors, the signal response
and the noise response. The mean of the quantiser output increases with increased bandwidth, shown in (3.54), as more of the signal is admitted into the correlator. As this bandwidth is increased, the noise incident on the quantiser becomes whiter, and the effective noise-floor reduction (depicted in Figure 3.13), caused by the quantiser non-linearity, is reduced. As the bandwidth of the filter approaches Fs {2, noise power which has been ‘whitened’ and spread out of the pass band, is aliased back
into the band of the received signal, and so contributes to the correlator output variance. B 1 2 3
Fc (MHz) 1.764 or 3.236 1.977 or 3.023 2.45 or 2.55
Bf (MHz) 1.547 1.760 3.893
Table 3.2: Optimum Filter Parameters (Fs = 10 MHz).
Theoretical results for two- and three-bit quantisers, depicted in Figures 3.20 and 3.21, respectively, exhibit similar loss surfaces. Optimum filter design parameters for one-, two- and three-bit quantisers are shown in Table 3.2 wherein both optimum
center frequencies, which lie symmetrically about Fs {4, are presented. The optimum
bandwidth increases with increased quantiser resolution as the quantiser transfer 83
4
Chapter 3: Quantisation and Filtering Losses
−1
L (dB)
−1.2 −1.4 −1.6 −1.8 −2 2
4
6
8
10
12
2
4
6
8
10
Fc (MHz)
Bf (MHz)
Figure 3.22: Loss Surface for One Bit Quantisation GPS C/A (Fs = 25 MHz)
characteristic, (3.43), becomes more linear. In the case of three-bit quantisation, the optimum value approaches the maximum allowable bandwidth, where the gain brought about by admitting more signal begins to outweigh the gain of ‘whitening’ the noise. Results of further theoretical analysis using a higher sample rate (Fs = 25 MHz) are depicted in Figures 3.22, 3.23 and 3.24. Here, the maximum allowable bandwidth is far greater than the significant bandwidth (for example, the bandwidth within which 90% of the signal power is contained) of the received signal. Optimum filter design parameters for one-, two- and three-bit quantisers are shown in Table 3.3. In a similar fashion to the 10 MHz case, the one- and two-bit loss surfaces increase to a maximum within 2 MHz and 4 MHz of front-end bandwidth, respectively, before smoothly decreasing as the bandwidth approaches Fs {2. The loss surface declines
less rapidly with increased bandwidth in the two-bit case, as compared to the one-bit case, due the quantiser’s ‘more linear’ response.
Examining Figure 3.24, it can be seen that the loss surface increases almost monotonically with increased bandwidth and is almost constant with changing centre frequency. This closely resembles classical filtering loss theory developed for linear analogue receivers [91], which is to be expected with a high number of quantisation levels and a high sample rate. Although this analysis focuses primarily on the GPS L1 C/A signal, the model described in Section 3.4.4 is applicable to a variety of similar DSSS signals. As an illustration of this, the GIOVE A E1-B/C and GIOVE B E1-B/C signals [89] were examined using sample rates of 25 MHz and 50 MHz with one-bit quantisation. It 84
12
Section 3.5: Front-end Filter Optimisation
−0.4
L (dB)
−0.6 −0.8 −1 −1.2 −1.4 12
10
8
6
4
2
2
Fc (MHz)
4
12
10
8
6 Bf (MHz)
Figure 3.23: Loss Surface for Two Bit Quantisation GPS C/A (Fs = 25 MHz)
−0.2 −0.4 L (dB)
−0.6 −0.8 −1 −1.2 10
8
6
4
2
2
4
6
8
Bf (MHz)
Fc (MHz)
Figure 3.24: Loss Surface for Three Bit Quantisation GPS C/A (Fs = 25 MHz)
85
10
Chapter 3: Quantisation and Filtering Losses B 1 2 3
Fc pM Hz q 3.361 or 9.139 4.942 or 7.558 4.645 or 7.855
Bf pM Hz q 1.870 3.653 8.290
Table 3.3: Optimum Filter Parameters (Fs = 25 MHz).
was found that the loss characteristics are similar to those of the GPS C/A signal, as depicted for the 25 MHz case in Figures 3.25 and 3.26. Relative to the GPS case, the optimum design parameters, listed in Table 3.4, evaluate to approximately twice the bandwidth. This is due to the BOC spectrum having two main lobes, rather than the single main lobe of the GPS, BPSK spectrum. A brief summary of these conclusions of this front-end optimisation is presented in [27].
L (dB)
−1.6 −1.8 −2 −2.2 2
4
6
8
10
12
2
Bf (MHz)
4
6
8
10
Fc (MHz)
Figure 3.25: Loss Surface for One-Bit Quantisation GIOVE-A E1 B/C (Fs = 25 MHz).
Signal GIOVE-A E1 B/C GIOVE-B E1 B/C
Fs (MHz) 25.0 50.0 25.0 50.0
Fc (MHz) 4.511 (7.989) 4.511 (7.989) 9.874 (15.130) 8.725 (16.270)
Bf (MHz) 3.447 3.447 3.707 7.223
Table 3.4: Optimum Filter Parameters for GIOVE E1-B/C Signals
86
12
Section 3.6: Model Validation
L (dB)
−2 −2.2 −2.4 −2.6 2
4
6
8
10
2
6
4
8
10
Fc (MHz)
Bf (MHz)
Figure 3.26: Loss Surface for One-Bit Quantisation GIOVE-B E1 B/C (Fs = 25 MHz).
3.6
Model Validation
3.6.1
Simulation Results
Validation, through simulation, of the theoretical results presented in Section 3.5 is presented here. Taking the loss surface of Figure 3.22 as an example, a range of digital filters were designed and implemented in a receiver simulation environment to investigate this approximation. Simulated wideband IF samples of a GPS L1 C/A signal were applied to the IF filter and, subsequently, quantised and correlated with a synchronised carrier and code replica. SNRc was then estimated over simulation time of 100 s. A total of 120 filters were designed and implemented, at points along the most prominent features of Figure 3.22. Forty filters used the optimal bandwidth (1.552 MHz) and a range of center frequencies and forty were placed along each of two lines between the optimal points (see Table 3.3) and the maximum bandwidth of 12.5 MHz. The filters used were eighth order Elliptic filters having a pass-band ripple of 0.1 dB, a stop-band attenuation of
20 dB and a 3 dB bandwidth of Bf .
Figure 3.27 shows the results of the simulation test points overlaid on the theoretical loss surface of Figure 3.22. The theoretical and simulation results have maximum and minimum percentage deviations of 4.94% and
1.37%, respectively,
with mean and standard deviations of 1.74% and 1.19%, respectively. This close agreement justifies the approximation of a real filter response with that of a ‘BrickWall’ for the purposes of loss estimation. 87
Chapter 3: Quantisation and Filtering Losses
−1.1 Simulation Theory
L (dB)
−1.3 −1.5 −1.7 −1.9 −2.1 2
4
6
8
10
12
2
4
6
8
10
Fc (MHz)
Bf (MHz)
Figure 3.27: One-bit Loss Surface: Simulation vs Theory (Fs = 25 MHz).
3.6.2
Real Signal Validation
A qualitative validation of this theory was conducted using a recording of the GPS L1 C/A signal. This data was collected from a rooftop environment under good signal conditions, with six high elevation satellites in line-of-sight, and no significant multipath component. A National Instruments front-end(10) was used, configured for a 25 MHz sample rate, a 6.42 MHz center frequency and a 14-bit signal representation. Prior to digitisation, the signal was passed through a wide anti-alias filter which limited the signal bandwidth to approximately 10 MHz. While this filter has a linear phase response and is sufficiently wide-band, it exhibits poor roll-off at low frequencies. This digitised signal was re-filtered by a range of filters and, subsequently, requantised at one-bit resolution. Through this re-filtering and re-quantisation, a range of front-end characteristics are impressed upon the signal, while preserving such real signal propagation effects as thermal noise, signal Doppler, electromagnetic and co-channel interference and ionospheric and tropospheric effects. In all, 28 filters were applied to the signal, similar in design to those described in Section 3.6.1, having a centre frequency of 6.42 MHz and 3 dB bandwidths ranging from 1 to 8 MHz. The filtering operation was carried out in software using 32 bit floating point precision. For each of the filtered data sets, a single satellite signal was acquired and then tracked using a 0.2 Hz, first order delay-lock-loop aided by a third order, 5 Hz phase(10)
The National Instruments [53] front-end consisted of a ‘NI PXIe-1065’ chassis containing a two-stage 3 GHz, ‘NI PXI-5690’ LNA, a 20 MHz bandwidth, ‘NI PXI-5600’ downconverter and a 14-bit, ‘NI PXI-5142’ digitiser and digital downconverter.
88
12
Section 3.6: Model Validation
48.2 48
Estimated C/N0 (dB Hz)
47.8 47.6 47.4 47.2 47 46.8 Measured Joint Theory Classical Theory
46.6 46.4 1
2
3
4 5 Bf (MHz)
6
7
8
Figure 3.28: C {N0 Vs Front-End Bandwidth using a real GPS signal.
lock-loop [104]. Once the tracking loops had locked and achieved bit-syncronisation, they were allowed to settle for a further period of twenty seconds, at which point a standard C {N0 estimator [91] was applied to the correlator outputs. This C {N0
estimate was averaged over the remaining 180 s of the data set.
Figure 3.28 shows a plot of the average C {N0 estimate versus Bf , along with
theoretical estimates of C {N0 . The theoretical curve, labeled ‘Joint Theory’, is the
sum of a C {N0 of 49.43 dB Hz, prior to filtering and quantisation, and the processing
loss shown in Figure 3.27. Given a fixed centre frequency, Figure 3.28 is equivalent to a vertical cross-section through the mesh depicted in Figure 3.27, along the plane Fc
6.42 MHz. The theoretical and real signal curves agree well, having maximum
values at 3.121 MHz and 3.333 MHz, respectively and trending in a similar fashion
with changes of Bf about these points. A disparity between the curves is evident, however, for large values of Bf . This is due to the assumption that the data set used is white and that the front-end anti-aliasing filter is sufficiently wide. As mentioned 89
Chapter 3: Quantisation and Filtering Losses above, this filter has a poor low-frequency roll off. As a result, when a wide (5 - 8 MHz) filter is applied in the re-filtering stage, the net bandwidth is less than expected. This reduced bandwidth causes an increase in the signal C {N0 in this
region.
Also depicted in Figure 3.28 is the curve labeled ‘Classical Theory’. This curve
represents an estimate of the C {N0 using the classical approach to loss analysis,
which addresses the losses incurred through filtering and quantisation independently. Traditionally, the effects of quantisation and filtering have been treated in isolation. The total loss has been assumed to be the product of one loss coefficient representing the loss due to filtering, similar to Section 3.3, and one loss coefficient representing the loss due to quantisation, neglecting the front-end filter (i.e. assuming hf rk s
δk ). The discrepancy between the two theoretical curves is significant. Moreover, not only does this curve exhibit a -1 dB bias, it suggests a trend of increasing C {N0
with increasing Bf which, as shown by the measured data and joint theory presented here, is not the case.
3.7
Conclusions
This chapter has presented a detailed description of the IF signal processing of a GNSS signal, considering the receiver operations applied to the received signal as it propagates from the LNA to the signal at the output of the DMF (or correlator). Intermediate operations such as filtering, quantisation, carrier and code demodulation and accumulation, have been examined. The performance of the DMF has been examined in three distinct cases: the ideal case, for which the received signal endures neither filtering nor quantisation; the filtering only case; and the case where both the effects of filtering and of quantisation are considered. The ideal case has provided an insight into the operation, capability and limitations of the DMF and has served as a benchmark against which the non-ideal receiver implementations have been compared. In particular, a loss coefficient, L, representing the effective degradation in SNRc , for a given receiver configuration, has been defined. An analysis of the effects of the front-end filer has been conducted which has concluded that these effects influence both the propagation of the signal mean to the correlator output, and, also, the propagation of the noise. When analysing the implications of quantisation on the performance of the DMF, the effects of front-end filtering have also been considered. In the joint examination of the losses incurred through both filtering and quantisation using a symmetric quantiser it has been illustrated that the total loss is a function of: the number of quantisation levels, the quantiser gain (Ag ), both the front-end filter bandwidth and its centre frequency and the receiver sample rate. This joint analysis has provided insight into the key features of the receiver design which influence the related processing loss. These results have been verified through the use of real GPS L1 C/A signals. 90
Section 3.7: Conclusions Utilizing this joint theory, an optimisation of the choice of filter design parameters: centre frequency and bandwidth, and the number of quantisation levels, which minimise the processing loss of a receiver, has been conducted. Novel results have been presented regarding the relationship between front-end filter bandwidth and processing loss for quantised signals. In the case of one- and two-bit quantisation, these new results are, in fact, contrary to the classical theory. In particular, it has been shown that an increase in bandwidth, for a given sample rate and quantiser resolution, not only fails to increase the effective SNRc , but can actually reduce it. A number of specific cases have been examined, including a selection of GNSS modulation formats and receiver configurations and the corresponding optimal receiver design parameters have been tabulated.
As discussed in Chapter 2, the correlator output, y rms, represents the fundamen-
tal baseband measurement from which all of the receiver estimates, of the received
signal parameters, are derived. It is clear, therefore, that to fully understand and quantify the performance of receiver baseband operations, this receiver measurement must first be analysed. This chapter has offered such an analysis and, having considered the performance of the DMF, Chapter 4 will proceed to consider some baseband operations. As the focus of this thesis is carrier tracking algorithms, Chapter 4 will consider what is, typically, the first tracking algorithm applied to a received signal, namely, the FLL.
91
92
Chapter 4
Analysis and Design of Frequency Lock Loops The previous chapter presented an analysis and optimisation of the GNSS DMF which is employed to extract the desired satellite signal from the ensemble of satellite signals, thermal noise and various interferences. This chapter will consider the formation of estimates of some signal parameters. Specifically, the process of estimating the frequency of the carrier and, in some cases, the dynamics of the carrier frequency, will be examined. As discussed in Chapter 2, to demodulate the GNSS signal the receiver must (at least) synchronise the frequency of the local carrier replica (used in the DMF) with the received satellite carrier frequency. The discrete update FLL is a recursive estimator of carrier frequency and is an attractive solution to the problem of carrier frequency synchronisation. This chapter considers four main aspects of the FLL: the design of loop filters for FLLs, the transient performance of the FLL in the presence of thermal noise, the steady state tracking error of the FLL in the presence of thermal noise and, finally, the choice of frequency discriminator. Novel design expressions for loop filter gains, specific to discrete update FLLs, are presented for first and second order (both critically damped and standard underdamped) loops in Section 4.2. A thorough analysis of four popular carrier frequency discriminators is presented in Section 4.3. This analysis is employed in Section 4.4, wherein new insight into the impact of thermal noise on the transient performance of the FLL is provided, along with a method of alleviating some of these effects. Novel closed form expressions for the noise bandwidth and for the thermal noise induced tracking error variance of first and second order FLLs are presented and the behaviour of the FLL across a wide range of signal strength conditions is examined in Section 4.5. Based on this thorough examination of the FLL, novel design considerations are presented which can, under certain circumstances, provide up to a 10 dB improvement in performance, over a ‘na¨ıve’ FLL design. 93
Chapter 4: Analysis and Design of Frequency Lock Loops
4.1
Problem Statement and System Architecture
This chapter considers the process of synchronisation of the receiver’s local carrier replica with a received GNSS signal, defined in the previous chapter and repeated here for convenience: rIF rk s sIF rk s
sIF rk s
?
n rk s
2P cos pωc kTs
θq c pkTs τ q d pkTs τ q .
(4.1)
As rIF rk s is ‘buried’ in thermal noise, to reliably estimate the received signal carrier
frequency the receiver observes the correlator values: y rms at a rate of 1{TL . Typically, TL will be equal to, or an integer multiple of, the spreading code period. Note
that the distinction between TI , in Chapter 3, and TL , in this chapter, is purely notational and stems from receiver implementation considerations. Generally, a receiver will compute correlator values over the shortest possible period (e.g. TI = 1 ms for GPS L1 C/A and TI = 4 ms for Galileo GIOVE A/B [61]), and then co-
herently sum these correlator values to produce correlator values at a rate of 1{TL .
Mathematically, of course, this is equivalent to correlation over TL seconds, and so the statistics of y rms can be expressed in terms of TL .
As has been shown in Chapter 3, the effects of quantisation and filtering on
the signal-to-noise ratio of y rms can be embodied in one loss coefficient, L. To
simplify this analysis, therefore, the ideal case of no font-end filtering and no quantisation, will be considered here. The effect of quantisation and filtering can then be encapsulated by scaling the coherent signal-to-noise ratio by L.
The real and imaginary parts of y rk s, in the ideal case, are denoted by: Im
Qm
< ty rmsu
? δωc TI P sinc 2 Rcode pδτ q cos δθ
= ty rmsu
? δωc TI P sinc 2 Rcode pδτ q sin δθ
δωc TI 2 δωc TI 2
ni rms
(4.2)
nq rms .
(4.3)
It is assumed here that, in the case where TL is not equal to one code period and that no external data-aiding is employed, the receiver has achieved bit synchronisation, such that the period TL does not ‘straddle’ a bit boundary. Furthermore, since only the carrier tracking loops are of interest here, it is assumed that the code tracking loop is ideal, and that the mean code tracking error is zero (i.e. τm
and Rpτm q 1).
0
The FLL tracks the carrier frequency by computing an estimate, e, of the difference in frequency between the received signal and the local carrier signal from the series Im and Qm . This estimate, e, is filtered by the loop filter and the filtered 94
Section 4.1: Problem Statement and System Architecture
Σ s[k]
Im
Sum and Dump
Σ
c[k]
Frequency Discriminator
Qm e
From Code Tracking Loop
OSC
Loop Filter
sin
cos
NCO
Figure 4.1: Block Diagram of a typical digital frequency lock loop.
estimate is used to control the receiver’s NCO. A simplified block diagram of this operation is depicted in Figure 4.1.
4.1.1
Linear Model of the FLL
Frequency discriminators, in GNSS receivers, are functions which produce estimates, e, of the carrier frequency error, δω. One estimation method examines the difference in power between a correlator pair, Im and Qm , which use a slightly high Doppler frequency estimate, and a correlator pair which use a slightly low Doppler frequency estimate. The difference in power can be used as a means of computing the carrier frequency error. This approach is discussed in detail in [60]. Another approach is to compute e from two or more adjacent I, Q pairs (Im , Qm ,Im1 , Qm1 ,...). This
approach estimates the phase error at each epoch, and estimates the frequency error as the average change in phase per integration period, TL . Generally, frequency discriminators can produce accurate estimates only over a limited range of δω, known as its linear region, which is inversely proportional to TL . The latter method of frequency estimation, known as the differential-phase technique, is by far the more popular and is the only method considered in this chapter. A carrier frequency estimate made using thermal noise corrupted samples, em , can, under certain conditions (which will be discussed in Section 4.3), be approximated by a gain plus an independent corrupting noise: em
KT D pθm θm1q L
nωm ,
(4.4)
where KD is a gain dependent on both the discriminator function and SNRc and nω is a corrupting noise. The distribution and correlation properties of nω are discussed in Section 4.1.2 and, in more detail, in Section 4.3. Other than the discriminator function, the FLL, depicted in Figure 4.1, is linear and can be represented by a system of z-domain transfer functions, where the update 95
Chapter 4: Analysis and Design of Frequency Lock Loops
nω
θ
+ _
δθ
δω c KD
D(z)
+
+
e F(z)
θˆ NCO(z) Figure 4.2: Linearised FLL Model
rate of the system is 1{TL . A linearised loop model can be quite useful as it facilitates the estimation of loop stability and tracking performance. Of interest here are the
transfer functions between the frequency estimate noise, nω , and the tracking error,
δω, denoted Hn pz q, and between the carrier frequency, ω, and the carrier frequency estimate, ω ˆ , denoted Hω pz q. These quantities are depicted in a linearised loop model
in Figure 4.2. These transfer functions can be expressed in terms of the loop filter, F pz q, the numerically controlled oscillator, N COpz q, and the discriminator function, Dpz q, as:
Hn pz q Hω pz q
∆Ωpz q Nω pz q ˆ pz q Ω
Dpz qF pz qN COpz q KD Dpz qF pz qN COpz q
1
KD Dpz qF pz qN COpz q , Ω pz q 1 K Dpz qF pz qN COpz q
(4.5) (4.6)
D
where upper-case symbols represent the z-transform of the corresponding lower-case time series. The transfer functions have similar spectral characteristics, differing only in magnitude (by a factor KD ). The functions F pz q and N COpz q represent the
z-transform of the loop filter and the numerically controlled oscillator, respectively. The scalar KD represents the gain of the discriminator. Further details on the value of this gain and the implications of its value are discussed in Sections 4.3 and 4.4. The numerically controlled oscillator, which is, essentially, a digital integrator, is defined as: N COpz q
zTL , z1
(4.7)
and, from (4.4), the z-transform of the discriminator function, Dpz q, is given by: D pz q
z1 . zTL
96
(4.8)
Section 4.1: Problem Statement and System Architecture
4.1.2
Noise Equivalent Bandwidth and Tracking Error Variance
The (two-sided) noise bandwidth, Bω , of the system Hω psq is defined as: Bω
1 2πTL
»π
π
|Hω peiω q|2dω.
(4.9)
Although this metric does not provide information about the noise performance of the system, it provides an intuitive insight into the behaviour, in particular the dynamic response, of the system. Generally, a system with a higher bandwidth will have a faster transient response and a smaller steady state error. Typically, the noise equivalent bandwidth and the loop order are specified in the loop design. Here, in Section 4.4, Bω is used as a metric to standardise the comparison of the performance of different carrier frequency discriminators. It is worth noting the presence of the discriminator gain, KD , in both the numerator and denominator of Hω pz q, in (4.6).
It will be shown in Section 4.3 that KD is a function of both the carrier frequency discriminator choice and of SNRc . This implies that Bω is not constant, but, rather, is dependent upon SNRc . It is, therefore, important to understand how KD varies with SNRc . Once KD is known the true value of Bω can be estimated and, moreover, the loop filter can be appropriately modified to restore Bω to its design value. This will be discussed in more detail in Section 4.4. To evaluate the thermal noise induced frequency tracking error, the spectral density of the measurement noise, nω , must be found. Examining (4.4), it can be seen that the frequency estimate is derived from the difference of two phase estimates, scaled by the intervening delay. Each phase estimate is produced from only one correlator pair, Im and Qm , and so the noise distorting these phase estimates is white. Being derived from the difference of two adjacent samples of white noise, the autocorrelation function of nω , denoted Rn rms, can be readily inferred:
R rms n
$ ' ' Rn ' & 0
Rn
1 ' ' ' %0
for m 0
for |m| 1 ,
(4.10)
otherwise
the Fourier transform of which yields the power spectral density (PSD), S n pwq, of nω :
S n pwq TL R0n
2TL R1n cospω q.
(4.11) (4.12)
This power spectral density can be used in conjunction with (4.5) to find the 2 , is thermal noise induced frequency error variance. The variance of δω, denoted σδω
97
Chapter 4: Analysis and Design of Frequency Lock Loops given by:
2 σδω
1 2πTL
»π
π
S n pwq|Hn peiω q|2 dω.
(4.13) (4.14)
The term R0n represents the total power in the discriminator noise whilst R1n represents the correlatedness of the noise, given a one loop update period offset. As will be shown in Section 4.3, R1n is negative and ranges in value from
R0n{2 to
0. Its exact value depends on both the carrier discriminator function used and the prevailing SNRc . Under high SNRc conditions, when the discriminator function behaves in a linear fashion, it can be seen from (4.4) that R1n
R0n{2 (as shown
in Section 4.3). This implies that S n pwq, for high SNRc conditions, is a high-pass
process. The loop filter and, thus, the system defined by Hn pz q, is low-pass and so
there exists, under these conditions, a large spectral separation between the system and the discriminator noise. As the value of SNRc is reduced, however, the value
of R1n approaches zero and S n pwq becomes more white, overlapping with the pass
band of Hn pz q. This results in a significant degradation in tracking performance. It
is critical, therefore, that the relationship between SNRc and R0n and R1n is clearly
2 . understood, such that the FLL can be designed to minimise σδω
It is worth emphasising here that the noise equivalent bandwidth, Bω , and the 2 , are quite different. The bandwidth, B , is a function tracking error variance, σδω ω
2 is a function of both the of the loop transfer function, Hω psq, only, whereas σδω
noise-related loop transfer function, Hn psq, and the discriminator noise spectrum,
S n pwq. For this reason, they cannot be directly related and, as will be shown in Section 4.5, behave very differently under low SNRc conditions.
The closed loop transfer function, Hω pz q, is manipulated through the choice of
the loop filter, F pz q. Generally, closed loop poles of Hω pz q are placed to achieve
a specified dynamic response. It is common, although perhaps not optimal, to use either a first order (proportional) or a second order (proportional and integral (PI)) style controller [61]. Such a controller is defined in (4.15), where A0 and A1 represent the proportional and integral gains, respectively, and the leading integrator is included to compliment the effect of the differentiator in Dpz q: F pz q
zTL z1
A0
zTL A1 z1
.
(4.15)
2 can be Assuming this form of controller, closed form expressions for Bω and σδω
found. By noting that: 98
Section 4.2: Design Equations for The Loop Filter
»π
|Gpeiω q|2dω
π
¾
Gpz qGpz 1 qz 1 dz,
Γ
(4.16) where Γ represents the anti-clockwise closed contour around the unit circle centred at the origin of the complex plane, the integrals (4.9) and (4.13) can be evaluated via the Cauchy Residue Method [81, 104]. For the first order loop A0
¡ 0 and A1 0.
Substituting (4.15) into (4.9), the
noise equivalent bandwidth, Bω , is readily found to be: Bω
2 AA0KKD T 0
(4.17)
.
D L
Applying the same approach to (4.13), the tracking error variance for a first order loop is given by: 2 σδω
n n 1 pA0 KD TL 1qq A0TLpRK0 p22R . A K T q 0
D
For the second order loop A0
(4.18)
D L
¡ 0 and A1 ¡ 0.
In a similar fashion, the noise
equivalent bandwidth is found to be: Bω
1 TL
2p2A0 A1 TL q A0 p4 KD TL p2A0 A1 TL qq
1
(4.19)
and the tracking error variance is given by:
2 σδω
n pR0
2R1n qTL 2A20 KD A1 p2 A0 KD TL q 2 p4 K T p2A A0 KD A1 TL qq 0 D L
n 2 2 2R1 KD TL AA1KT2L p4 A0KKDTpAp02A A1TALqpT2Aqq0 0 D 0 1 L D L
A1 TL q
.
(4.20)
2 tend to be verbose For higher order loops, closed form solutions for Bω and σδω
and relatively obtuse and, consequently, it is often easier to resort to numerical integration techniques. Also, considering the typical dynamics of the carrier frequency of GNSS signals, a second order FLL is, generally, sufficient in practise. For these reasons, only the first and second order FLL will be considered in this chapter.
4.2
Design Equations for The Loop Filter
Assuming the loop filter form defined in (4.15), it is useful to be able to calculate the filter coefficients (i.e. A0 and A1 ) for a given Bω . For the first order loop there is only one degree of freedom, A0 , and so (4.17) can be easily manipulated to yield: 99
Chapter 4: Analysis and Design of Frequency Lock Loops
A0
K p12BωB T q . ω L D
(4.21)
The second order loop, in contrast, poses a more difficult problem. There are two degrees of freedom, corresponding to the two poles of (4.5). These poles can be represented by za
eβ p1
η
q and zb
eβ p1ηq , where β is the pole decay-rate
parameter and η represents the pole damping parameter [104]. η has a significant impact on the transient response of the loop and is, therefore, a critical design parameter. β, in contrast, is only of interest insofar as it dictates Bω . It is possible, therefore, to choose Bω and η as two design parameters in terms of which A0 and A1 can be expressed, thereby leaving β as a dependent variable. Simplifying (4.6), the closed loop system, Hω pz q, reduces to: Hω pz q z2
KD A0TL KD zA0TL K D zA1TL2 z 2 KD A0 TL KD A1 TL2 KD A0 TL
1
(4.22)
and has poles:
za
12
1
α
a
4 pKD A0 TL 1q
a
2 α 4 pKD A0TL 1q α 2 KD A0 TL KD A1 TL2 .
zb
Equating these poles to eβ p1
η
(4.23)
α2
(4.24)
α2
(4.25)
q and eβ p1ηq , the following expressions for A0 and
A1 in terms of β and η can be found:
A0 A1
2β
1K eT
D L e β1 η
p
(4.26)
q
1
ηq
eβ p1 KD TL2
1
eβ p1
ηq
.
(4.27)
At this point, it is possible to find the filter coefficients directly, given desired pole positions. Typically, however, the exact pole positions are not known to a designer. Rather, the damping contour (the fixed η value) along which the poles should lie is known and β is a dependent variable. Both the loop update rate, 1{TL , and the
noise bandwidth, Bω , are known and so, given an expression for β in terms of Bω and TL , the filter gains could be found.
Given the expression for the noise bandwidth, presented in Section 4.1.2 and repeated here for convenience, and using expressions (4.26) and (4.27), Bω , TL and 100
Section 4.2: Design Equations for The Loop Filter β can be related:
Bω TL
A1 TL q A p4 2Kp2AT0 p2A 1 A1 TL qq 0 0 D L 2 eβ eβηcothepββpq2 ηq1 eβ 2βη
β
e
3e
e 2e p
3β
βη
β 2 η
q
(4.28)
5e p
β 4 η
q 3eβ p3
2η
q
β 2βη
e
. (4.29)
Ideally, (4.29) would be manipulated to yield an expression for β in terms of Bω , TL and η. The resulting expression could then be substituted into (4.26) and (4.27) to find the desired filter coefficients. Unfortunately, to the best of the authors’s knowledge, no such expression is currently attainable. To proceed, some generality must be sacrificed. To simplify the problem, specific η values can be considered. Two commonly used values are: η
0, which gives a critically damped response and η i, which
gives a standard underdamped response. Taking the critically damped case first, substitution of (4.26) and (4.27) into (4.19) and setting η Bω TL
η 0
1
eβ
1
p1
eβ 4
eβ q
0 yields:
5eβ
(4.30)
.
3
This expression has three solutions for β, two of which are complex in value, and one of which is real valued. By definition, β is real and so:
β
ln
21{3 κ2 κ p1
κ 203
234Bω TL
3 pBω TL 5q
a
75
3Bω TL q 22{3 p23 3κ p5 Bω TL q
21Bω TL q
(4.31)
27pBω TL q2 156Bω TL
81pBω TL q
2
1{3
.
(4.32)
This expression for β is rather cumbersome but can be well approximated by the quadratic β
c1Bω TL c2pBω TLq2 where the coefficients c1 π{4 and c2 1{6
have been found through a numerical least squares fit and provide the more useful expressions for the loop filter coefficients (4.33) and (4.34). Substitution of this approximate function, and η
0, into (4.26) and (4.27) yields the following expressions
for the coefficients of a second order critically damped FLL filter: 101
Chapter 4: Analysis and Design of Frequency Lock Loops
1 e 6 Bω TL p2Bω TL 3πq A0 KD TL 1
1
A1
e 2 Bω TL π e 6 pBω TL q 1
2
e
(4.33) 1 B T π 4 ω L
2
(4.34)
.
KD TL2
These expressions for A0 and A1 have been numerically optimised for Bω TL in the range 0.0 to 0.5 (which is most commonly the case in practise) and are accurate
to within 1.8% in this region. The approximation holds for Bω TL values up to 0.9, but the error increases to 2.5% at Bω TL = 0.75 and to 4.3% at Bω TL = 0.9.
Note, however, that to design a critically damped loop using the exact expressions developed here, a designer, given a desired loop bandwidth, Bω , and chosen loop update period, TL , can use (4.31) to find the necessary decay rate parameter which, in turn, can be used, in conjunction with η required loop filter coefficients.
0, in (4.26) and (4.27) to find the
In a similar fashion to the critically damped case, the standard underdamped loop can be examined. In the standard underdamped case the value of η is given
1 and corresponds to the value ζ ?12 in the traditional notation of a continuous time system [104]. Substituting η i into (4.29) and simplifying yields: by η 2
Bω TL
4pcospβ q 1 coshpβ qq
η i
coshpβ qp3 cothpβ q
8q
cschpβ q
4 cospβ q cothpβ q 8 cospβ q
9 sinhpβ q .
(4.35)
Unlike (4.30), to the best of the author’s knowledge, no closed form solution of (4.35) for β currently exists and so a numerical approximation via a quadratic is used. The quadratic chosen is, again, β are: c1
0.6534 and c2 1{5.
c1Bω TL c2pBω TLq2, where the coefficients
As in the critically damped case, (4.36) and (4.37)
have been numerically optimised for Bω TL in the range 0.0 to 0.5 and, in this case, are accurate to within
2.0% in this region.
The approximation holds for
¤ 0.9, but the error increases to 3.2% at Bω TL = 0.75 and to 6.1% at Bω TL = 0.9. Substitution of this approximate function and η i into (4.26) and (4.27) Bω TL
yields expressions for the coefficients of a second order standard underdamped FLL filter: 102
Section 4.3: Discriminator Analysis
1K eT
2γ
A0 A1 γ
1
D L e2γ
(4.36)
2eγ cospγ q
KD TL2
51 Bω TL
Bω TL
41 4π
(4.37)
.
(4.38)
The significance of an accurately designed FLL loop is discussed in Section 4.4. In particular, in a weak signal environment (low SNRc ), it is important that the poles of the system can be accurately placed to minimise thermal noise induced tracking error whilst still maintaining a sufficiently fast transient response. Unlike the expressions presented in this section, traditional continuous-time approximations [61, 80] fail to provide this accuracy. When continuous time filter design equations are used for the design of a discrete update system (such as a GNSS receiver) the resultant noise bandwidth tends to be larger that the design value. The error generally becomes more prominent with increased Bω TL (for example, it exceeds 10% when Bω TL
¡
0.1). In light of this, (4.33), (4.34), (4.36) and (4.37) offer a marked improvement
in the achievable loop design accuracy. A summary of results presented here can be found in [31].
4.3
Discriminator Analysis
The treatment of FLL performance analysis thus far has assumed a general model for the carrier frequency discriminator. The discriminator characteristics have been embodied in the scalar KD , both in the analysis of noise equivalent bandwidth and in the autocorrelation values R0n and R1n for the analysis of thermal noise induced tracking error variance. Four popular carrier frequency discriminators are analysed in this section; the four-quadrant arctangent and arctangent discriminators [61, 80, 114] and the cross-product and decision-directed cross-product discriminators [61, 85, 86]. Expressions for KD , R0n and R1n are presented below for each discriminator considered here.
4.3.1
The Four-Quadrant Arctangent Discriminator
The four-quadrant arctangent discriminator is defined as: eAtan2 m
1 arctanpDot, Crossq TL
where 103
(4.39)
Chapter 4: Analysis and Design of Frequency Lock Loops
Dot Im Im1
Qm Qm1
Cross Qm Im1 Im Qm1 .
(4.40)
The frequency estimate is equal to the difference between two coherent phase estimates and is sensitive to data modulation. Consequently, care must be taken when applying it to data modulated signals. The response of this discriminator to frequency error is shown in Figure 4.3 for a range of SNRc values. It is evident that the response of this discriminator is sensitive to SNRc , since both the linear region of the discriminator function and the slope of the discriminator function change with changing SNRc . These effects have a significant impact on the performance of the FLL and will be discussed in more detail in Section 4.4. The discriminator gain is defined as the slope of the discriminator function when the frequency error is zero [61, 80]. Under strong signal conditions, when the SNRc is very large, this slope is approximately unity and, as the signal strength is reduced, the slope of the discriminator function reduces. For the arctangent based discriminator, examination of the distribution of the apparent phase of a signal yields an insight into the discriminator gain. If the correlator values, Im and Qm , are interpreted as a complex value Im jQm ,
then the apparent phase of the signal is given by =pIm
jQm q
δθrms
φrms,
where =pxq denotes the angle of x measured anti-clockwise from the positive real axis, δθrms is the instantaneous phase (as depicted in Figure 4.2) and φrms is the
excess phase induced by thermal noise, ni rms and nq rms. The excess phase has zero
mean, and is distributed according to the probability density function ppφq, given by [47]:
c SNRc 1 e 2 π SNRc 2 ppφq 1 cospφqe 2 SNRc cospφq 2π " 2 *
SNRc cospφq ? 1 erf , π ¤ φ ¤ π. 2
1s
(4.41)
Any given pair of consecutive phase samples can, thus, be written as: δθrm φrm
1s and δθrm 1s
δω rmsTL
φrms, where δω rmsTL represents the
phase advance per sample interval. Noting that the frequency estimate is simply a scaled measure of the phase advance per sample period, the notation that follows
has been simplified by normalising the frequency discriminator functions by TL . As the thermal noise samples, ni and nq , are independent, so too are samples of the apparent phase and, thus, the mean value of the carrier frequency estimate (normalised by TL ) is given by: 104
Section 4.3: Discriminator Analysis
3
Atan2
µe TL (rad)
2 1 0 −1 −2
Noise−Free
−3 −3
−2
−1
10 dB
0 δω TL (rad)
5 dB 1
0 dB
2
3
Figure 4.3: Four-quadrant arctangent discriminator response to frequency error for a noise-free signal and three different SNRc cases.
µe TL
»π »π
π π
f pδωTL
φ2 φ1 qppφ1 qppφ2 qdφ1 dφ2 ,
(4.42)
where f pxq represents the noise-free carrier frequency discriminator input-output
characteristic (as depicted in Figure 4.3). That is, f pxq represents the mapping of the
true phase advance per sample period to the discriminator’s noise free estimate. An
expression for f pxq, representing the four-quadrant arctangent function, can be easily
found by substituting Im and Qm1
cospδωTL
φ 2 q, Q m
sinpδωTL
sinpφ1q into (4.39) and scaling the result by TL.
φ2 q, Im1
cospφ1q
Alternatively, it can
be composed from inspection of the noise-free curve of Figure 4.3, whereby a more elementary expression can be found. Here, the following expression represents the four-quadrant arctangent discriminators:
f Atan2 pδωTL q δωTL upδωTL
π q upδωTL π q
(4.43)
where upxq is the Heaviside step function. Also in Figure 4.3, the mean discriminator output, as predicted by (4.42), is plotted for a selection of SNRc values. It can be
seen that both the discriminator’s linear region and its gain reduce with reducing SNRc . The value of KD forms a part of the closed loop transfer functions, (4.5) and (4.6), and so dictates the closed loop performance, from its transient response and steadystate errors to noise bandwidth. It is, therefore, quite an important performance metric. An expression for KD can be found by differentiating (4.42) with respect to δω and taking the limit as δω
Ñ 0.
By noting that the derivative of the Heaviside
step function is the Dirac delta function, δ pxq, that the distribution ppφq is even (i.e. ppφq ppφq) and using (4.43), it can be shown, after some simplification, that: 105
Chapter 4: Analysis and Design of Frequency Lock Loops
1 0.8
KD
0.6 0.4 Atan2 Theory Atan Theory Atan2 Simulation Atan Simulation
0.2 0 −15
−10
−5
0
5 10 SNRc (dB)
15
20
25
30
Figure 4.4: Theoretical and simulated discriminator gain curves for a range of typical SNRc values for both the arctangent and four-quadrant arctangent discriminator.
Atan2 KD
1 4π
»π
ppθqppθ π q dθ.
(4.44)
0
The discriminator gain curve, given by (4.44), and simulated results are plotted against SNRc in Figure 4.4. The variance of nω for the four-quadrant arctangent discriminator is easily estimated using a similar approach to that of (4.42). For the purposes of FLL performance analysis, it is reasonable to assume that the loop will be locked and that the
tracking error will be relatively small. The dependence of Var pnω q on δω can, thus,
be neglected and Var pnω q can be approximated by its value at δω Var pn
ω
q
1 TL2
»π »π
π π
0, via:
f pφ2 φ1 q2 ppφ1 qppφ2 qdφ1 dφ2 .
(4.45)
While (4.45) provides an accurate estimate of the variance of nω across the entire range of SNRc , it is interesting to consider the high SNRc , linear approximation. Examining (4.39), it can be seen that the frequency estimate is derived from the difference of two phase estimates, scaled by the intervening delay. When the SNRc is large (e.g. 20 dB) and the frequency error is small and constant, the distribution of the phase estimates is approximately Gaussian and the variance is approximately equal to 1{ SNRc . Thus, nω can be approximated as a Gaussian random variable, with variance given by:
Var peq Var pnω q
2 . SNRc TL2
(4.46)
Simulated and theoretical values of TL2 V arpnω q are plotted in Figure 4.5 for a 106
Section 4.3: Discriminator Analysis 2
10
1
c
10 T2L Var(nω) (rad2)
Linear Model Atan2 Simulation Atan Simulation Atan2 Theory Atan Theory
2/SNR
0
10
−1
10
−2
10
π2 3
π2 12
−3
10
−15
−10
−5
0
5 10 SNRc (dB)
15
20
25
30
Figure 4.5: Theoretical and simulated discriminator variance (V arpnω q) curves for a range of typical SNRc values for both the arctangent and four-quadrant arctangent discriminator and the linear model.
range of SNRc values, in addition to this linear approximation. It can be seen that for large SNRc values, the approximation is accurate. As SNRc reduces and approaches (approximately) 10 dB the variance trend begins to diverge from the linear model. At this point the distribution ppφq becomes less Gaussian in shape, and the linear
model fails. Firstly, V arpnω q exceeds the linear model estimate and, subsequently, as the discriminator saturates, the variance converges to a constant upper bound.
For very low SNRc values, the distribution of nω tends toward a uniform distribution over the intervals
rπ, πs.
The upper bound on V arpnω q can, assuming a uniform
distribution of nω , be easily estimated as
π2 3 .
The carrier frequency discriminators operate on two successive phase estimates
which are corrupted by white noise. The auto-correlation function, Rn rms, therefore,
is only non-zero for m 0 and |m| 1 and so the PSD, S n pω q, is completely defined
by the two values: Rn r0s and Rn r1s. The m
0 case is simply the variance of nω , and is given by (4.45). The autocorrelation function at |m| 1 can be found from the distribution, ppφq, and the discriminator function, f pxq. By noting that φ is white, the expectation of the product of two successive carrier frequency estimates, f pφ1 φ2 q and f pφ2 φ3 q, can be written as:
1 Rn r1s 2 f pφ1 φ2 qf pφ2 φ3 q TL
T12 L
»π »π »π
π π π
f pφ1 φ2 qf pφ2 φ3 qppφ1 qppφ2 qppφ3 q dφ1 dφ2 dφ3 .
107
(4.47)
Chapter 4: Analysis and Design of Frequency Lock Loops
0.5
−Rn1/Rn0
0.4 0.3 0.2 0.1 Atan2 Atan
0 −15
−10
−5
0
5 10 SNRc (dB)
15
20
25
30
Figure 4.6: The ratio R1n {R0n versus SNRc for the four-quadrant arctangent and arctangent discriminators.
Following similar reasoning to that producing (4.46), the autocorrelation value R1n
can be shown to be approximately equal to
ure 4.6 is a plot of
of SNRc
values(1) .
R1n R0n
12 R0n, for high SNRc values.
Fig-
for the four-quadrant arctangent discriminator, over a range
As expected, for high values of SNRc , the Rn rms curves agree
well with the linear model given by Rn r0s 2{ SNRc and Rn r1s 12 Rn r0s. As the value of SNRc is reduced and the nonlinearity of the discriminator becomes manifest, the correlation of nω decays. For very low values of SNRc , the correlatedness of nω tends toward zero and the noise becomes white, spreading across the PSD of the useful signal. The significance of the values derived here will be discussed in detail in Sections 4.4 and 4.5. It is worth noting that this discriminator derives frequency estimates from the difference of two coherent phase estimates. Each coherent frequency estimate is sensitive to data modulation. As a result, were the carrier modulated by a binary phase shift keyed (BPSK) data sequence (as is common in GNSS signals, for example GPS L1 C/A), this discriminator would produce sporadic, corrupted frequency estimates whenever the data bit changed sign. For this reason, this, and other coherent frequency discriminators, should only be used on unmodulated, pilot channels, or in conjunction with a data wipe-off procedure. Such data wipe-off procedures include the use of a previously received navigation message to remove the current data modulation and assisted GPS (A-GPS) procedures, whereby the current data message is relayed through another means (through a cellular network, for example)[61, 80]. The curves in Figure 4.6 were calculated using (4.47), (4.43) and (4.50) for f pxq, as appropriate. Quasi Monte Carlo numerical integration techniques were used. The integral becomes oscillatory for high values of SNRc (above 15 dB) and so the curves have been forced to converge to the linear-model value of 21 . At the time of writing, no closed form expression for (4.47) has been found. (1)
108
Section 4.3: Discriminator Analysis
µe TL (rad)
1 0
Atan −1 13 dB
Noise−Free −2
−3
−2
−1
0 δω TL (rad)
8 dB 1
3 dB
2
3
Figure 4.7: Arctangent discriminator response to frequency error for a noise-free signal and three different SNRc cases.
4.3.2
The Arctangent Discriminator
The arctangent discriminator is defined as:
eAtan m
T1 U W L
arctanpQm {Im q arctanpQm1 {Im1 q ,
(4.48)
where
U W px q
$ ' ' x ' &
π
x
' ' ' %x
π
for x ¥ π {2
for π {2 ¡ x ¥ π {2 for
(4.49)
π{2 ¡ x.
The function U W pxq is simply a phase-unwrapping function which maps the phase estimate to the interval
r π2 , π2 s and, thus, mitigates the effect of data modulation.
The response of this discriminator to frequency error is shown in Figure 4.7 for a range of SNRc values. It is evident that, like the four-quadrant arctangent discriminator, the response of this discriminator is sensitive to SNRc . For the arctangent discriminator, the impact of SNRc is more pronounced and its onset occurs at a higher value of SNRc , relative to the four-quadrant arctangent discriminator. This is, in part, due to the difference between the linear regions of the two discriminator functions. The mean value of the arctangent discriminator can be found using (4.42), using the following equivalent expression for the discriminator noise free characteristic: 109
Chapter 4: Analysis and Design of Frequency Lock Loops
f
Atan
pδωTLq 2π
δωTL π u TL δω
π 2
u TL δω
3π 2
u TL δω
3π 2
u TL δω
π 2 (4.50)
.
Using (4.50), differentiating (4.42) with respect to δω and taking the limit as
δω
Ñ 0 it can be shown that: Atan KD
1 2π
»π π 2
ppθqp θ
3π 2
dθ 2π
»π
p pθ qp θ
π 2
π dθ. 2
(4.51)
Atan , together with simulated results versus SNR , is illustrated in FigA plot of KD c
ure 4.4. The variance of nω for the arctangent discriminator is easily estimated using a similar approach to that of the four-quadrant arctangent discriminator (using
(4.45)). Simulated and theoretical values of V arpnω q are plotted in Figure 4.5 for
a range of SNRc values. It can be seen that, for large SNRc values, the linear approximation is accurate and, as SNRc reduces and approaches (approximately) 13 dB, the variance trend begins to diverge from the linear model. Similar to the four-quadrant arctangent trend, V arpnω q exceeds the linear model estimate and,
subsequently, the variance converges to a constant upper bound. For very low SNRc
values, the distribution of nω tends toward a uniform distribution over the interval
rπ{2, π{2s. The upper bound on V arpnω q is, thus, π12 . 2
Using (4.47) and (4.50), the value of R1n for the arctangent discriminator can be
estimated. Figure 4.6 is a plot of R1n for the arctangent discriminator, over a range Rn 0
of SNRc values. Again, the value of R1n for the arctangent discriminator follows a
similar trend to that of the four-quadrant arctangent discriminator, but converges to its low SNRc value (of zero) at a higher value of SNRc . Unlike the four-quadrant arctangent based discriminator, this discriminator mitigates the effect of data modulation on the frequency estimate by reducing the estimate of the phase increment to the interval
r π2 , π2 s
via U W pxq. Whilst this
discriminator has the advantage over the four-quadrant arctangent discriminator of being usable on BPSK modulated channels without the need for aiding (data wipeoff), it comes at the cost of reduced performance. This will be discussed further in Section 4.5.
4.3.3
The Cross Product Discriminator
The cross product discriminator is defined as [61, 114, 85, 86]: 110
Section 4.3: Discriminator Analysis
eCP m
QmIm1 ImQm1.
(4.52)
In contrast with the arctangent based discriminator functions, the cross product discriminator is, mathematically, readily described. Despite this, previous treatments of FLLs employing this discriminator [86, 85, 61, 80] have made some assumptions regarding the correlation function of this discriminator which, it will be shown here, are invalid. These assumptions have a significant impact on the accuracy of tracking performance estimates and need to be examined. The mean value of the discriminator functions, for a given frequency error, can be found by substituting (4.3) into (4.52) and calculating the expectation:
µe TL
E rQmIm1 ImQm1s P dmdm1 sincp δωm2 TL q sincp δωm21TL q sinpδθm δθm1q P dmdm1 sinc2
δωm TL 2
sinpδωTL q,
(4.53) (4.54)
1 and, assuming that the code tracking error is small, Rpτ rmsq 1, and pδθm δθm1 q δωTL . Of course, if this discriminator were used on a data modulated signal, the assumption dm dm1 1 would not hold, and the frequency estimate would be inverted each time dm dm1 . where we note that, for data-less signals, dm dm1
This would lead to significant performance degradation. It is also assumed that the
frequency error will not change by asignificant amount over one sample period, thus: δωm1 TL δωm TL δω T sincp q sincp q sinc2 m L . Examining (4.54), two significant fea2
2
2
tures become apparent. Firstly, µe TL is a function of the signal power, P . Unlike the
arctangent based discriminators, which are self-normalising, the frequency estimate produced by the cross-product discriminator must be normalised by an estimate of P . Typically, GNSS receivers maintain an estimate of P for a variety of different demodulation tasks, and so this normalisation does not pose a significant problem. Secondly, it is evident from (4.54) that µe TL is not noise dependent. The cross product discriminator is unique amongst the four discriminators in this respect. Differentiating (4.54) with respect to δω, taking the limit as δω
Ñ 0 and assum-
ing perfect signal power normalisation, the gain of the discriminator, KD , is found to be unity for all values of SNRc . The mean discriminator response to frequency error (which is independent of SNRc ) is shown in Figure 4.8. Also shown in Figure 4.8 is 2 δωm TL sinpδωTL q and it can be seen that the sinc term of (4.54) attenuates the 2 discriminator estimate.
The variance of the cross product discriminator can be estimated in a similar manner to (4.54). By substituting (4.3) into (4.52) and calculating the expectation of the square, after some simplification, it can be shown that: 111
Chapter 4: Analysis and Design of Frequency Lock Loops
1
sin( δω TL )
CP
µe TL (rad)
0.5 0 −0.5 −1 −3
−2
−1
0 δω TL (rad)
1
2
3
Figure 4.8: Cross product (CP) discriminator and sinpδωTL q response to frequency error (δωTL ).
σe2 TL2
E pQmIm1 ImQm1q2 pµeTLq2 N0 2T
L
2P 2 2
N0 TL
P
N0 2TL
1 SNR c
sinc2 N0 2TL
δωm TL 2
2
2
sinc2
δωm1 TL 2
2
(4.55) (4.56)
2 SNR , c
(4.57)
where (4.57) is found in the limit as δω by 1{P .
Ñ 0 and by assuming perfect normalisation
In a similar fashion, the autocorrelation value R1n can be found. Previous work [86] has assumed this value to be zero. This is not the case, however, since there exists a significant negative autocorrelation at one sample offset, as will be shown below. This correlation, and its dependence on SNRc , has a significant impact on tracking performance. This will be discussed further in Section 4.5. The value of R1n is given by:
R1n TL2
E pQmIm1 ImQm1qpQm1Im2 Im1Qm2q
δωm2 TL δωm TL P sinc sinc 2 2
δωm1 TL P sinc sinpδθm2 δθm1 q sinpδθm1 δθm q 2
N0 2T cospδθm1 δθmq . (4.58) L 2
Taking the limit of (4.58) as δω
Ñ 0, noting that when δω 0 then δθm δθm1 112
Section 4.3: Discriminator Analysis
0.5
−Rn1/Rn0
0.4 0.3 0.2 0.1 CP DDCP
0 −15
−10
Figure 4.9: The ratio
−5
0
5 10 SNRc (dB)
15
20
25
30
R1n {R0n versus SNRc for the cross-product discriminator.
δθm2 and assuming normalisation by 1{P , then (4.58) reduces to:
SNRc 1 . R the ratio R converges to 12 ,
(4.59)
R1n TL2
For high SNRc values,
n 1 n 0
as do the arctangent
based discriminators and the linear model presented in Section 4.3.1. As the value 2 of SNRc reduces, the term SNR c of (4.57) becomes dominant and the ratio
RR
n 1 n 0
decays to zero. The result of this is a whitening of the discriminator noise PSD. A plot of
RR
n 1 n 0
versus SNRc is shown in Figure 4.9.
This curve is more gradual than that of Figure 4.6, the value of
R1n{R0n begins
to significantly reduce from a value of 21 , with reducing SNRc , at a higher value of
SNRc than that of the arctangent based discriminators. This reduction in correlation occurs at a lower rate, however, and for SNRc values lower than (approximately) 4.5 dB the value
R1n{R0n is larger for the cross product discriminator than for the
four-quadrant arctangent discriminator. The implications of this fact are discussed further in Section 4.5.
4.3.4
The Decision Directed Cross-Product Discriminator
The decision directed cross product discriminator is the non-coherent equivalent of the cross-product discriminator and is defined as [61, 85, 86]:
eDDCP m
pQmIm1 ImQm1q sgn pIm Im1 Qm Qm1 q .
(4.60)
The independence of this frequency estimate to data modulation can be illustrated 113
Chapter 4: Analysis and Design of Frequency Lock Loops
Noise Free
µe TL (rad)
1
13 dB
5 dB
3 dB
0.5 0 −0.5 −1
−3
−2
−1
0 δω TL (rad)
1
2
3
Figure 4.10: Decision directed cross product discriminator response to frequency error for a noisefree signal and three different SNRc cases.
by noting that the term dm dm1 can be factored out of both pQm Im1 Im Qm1 q
and sgn pIm Im1
Qm Qm1 q and by noting that d2m
1 for a BPSK signal.
The
response of this discriminator to frequency error is shown in Figure 4.10 for the noise-free case and a range of SNRc values. The response for the noise-free case is similar to that of the cross-product discriminator but it also exhibits sign inversions at π, due to the sgn p q term in (4.60). Unlike the cross product discriminator, the
mean response is sensitive to SNRc .
It has been shown in Section 4.3.3 that the mean value of pQm Im1 Im Qm1 q
in (4.60) is independent of SNRc and, therefore, it is the sgn pIm Im1
Qm Qm1 q
factor in (4.60) that gives rise to the SNRc dependency of the discriminator mean. The attenuation of the mean of the discriminator arises for the sporadic occasions when, due to thermal noise, sgn pIm Im1
Qm Qm1 q dm dm1 .
A simple illustration of this effect is revealed by considering the case where the frequency error is constant. In such a case, the mean value of eDDCP would m
be equal to the average value of pQm Im1 Im Qm1 q over all the occasions when
sgn pIm Im1
Qm Qm1 q is equal to dm dm1 , minus the average value of the value
of pQm Im1 Im Qm1 q over all the occasions when sgn pIm Im1 equal to dm dm1 .
Qm Qm1 q is not
Since pQm Im1 Im Qm1 q represents the signal part of eDDCP , a simplified m
model for the discriminator can be assumed:
µe TL
E rQmIm1 ImQm1sKBooton,
(4.61)
where KBooton represents the Booton equivalent gain [84]. A Booton equivalent gain is a scalar which minimises the mean square error between the product of two random variables and the product of this scalar and one of the random variables: 114
Section 4.3: Discriminator Analysis
KBooton
D
argmin pab akq2 k
E
,
(4.62)
where a and b are the random variables and k is a scalar. It is shown in [84] that if a and b are independent, then the right hand side of (4.62) reduces to the expected value of b. Assuming, therefore, independence between pQm Im1 Im Qm1 q and sgn pIm Im1
Qm Qm1 q, then a na¨ıve model
for the mean of (4.60), presented in [86], is given by:
µe TL
E rQmIm1 ImQm1sPEqual E rQmIm1 ImQm1sPN otEqual E rQmIm1 ImQm1sp1 2PEqual q,
Qm Qm1 q is equal to dm dm1
where PEqual is the probability that sgn pIm Im1
and PN otEqual
(4.63)
1 PEqual . An expression for PEqual can be found by analogy with
the bit error rate of differential BPSK demodulation and is given by [94]: PEqual
12 e
1 2
SNRc
(4.64)
.
The Booton gain for (4.63) is, thus, given by p1 2PEqual q. Considering the
gain of this discriminator; it has been shown in Section 4.3.3 that the slope of
pQmIm1 ImQm1q is unity and so the gain KDDDCP
is equal to the Booton gain.
A plot of this approximation to the discriminator gain is shown in Figure 4.11 (broken line) along with simulated data (markers). It is clear that, while both the simulated data and this approximate gain exhibit a similar trend, there is a notable discrepancy between them. This discrepancy can result in erroneous performance estimates and must, therefore, be addressed. The cause of the disagreement between the model presented in (4.63) and the true performance (or, for example, the simulated results of Figure 4.11) is the assumption of the statistical independence of pQm Im1 Im Qm1 q and sgn pIm Im1
Qm Qm1 q.
These two variables share common terms (Im , Im1 , Qm and Qm1 ) and, of course, are not independent. An exact expression for the mean of the discriminator, and,
subsequently, its gain would involve treatment of both normal and product-normal distributions and the nonlinearity of the sgn p q operator, and so has not been at-
tempted here. Instead, an approximate expression for the discriminator gain has been found numerically. The following form was chosen:
1 ec c1 0.388,
DDCP KD
115
1
SNRc
(4.65)
Chapter 4: Analysis and Design of Frequency Lock Loops
1 0.8
KD
0.6 0.4 Simulated Data
0.2
1−e−0.5 SNRc 1−e−0.388 SNRc
0 −15
−10
−5
0
5 10 SNRc (dB)
15
20
25
30
Figure 4.11: Theoretical (solid line) and simulated (markers) discriminator gain curves for a range of typical SNRc values for the decision directed cross product discriminator.
where the constant, c1 , was found via a minimum least-squares fit (MLS) to the simulated data points of Figure 4.11. Noting that sgn pxq2
1, the variance of the decision directed cross product dis-
criminator can easily be shown to be equal to that of the cross product discriminator. The same cannot be said, however, for the the autocorrelation value R1n . While the
term pQm Im1 Im Qm1 q contributes to an increase in R1n with decreasing SNRc , the term sgn pIm Im1
Qm Qm1 q introduces a decorrelation between successive
samples. A plot of simulated R1n values versus SNRc is shown in Figure 4.12. The value of R1n initially follows a
TL2 SNRc 1 trend with decreasing SNRc (depicted
by the solid black line in Figure 4.12). As SNRc reaches approximately 10 dB, R1n
deviates from this trend, reaching a minimum value of approximately 0.2TL2 at 2.7 dB, before tending toward zero. The reduction in the magnitude of R1n is due to the reduction of the probability that two successive values of sgn pIm Im1
have the same sign. This trend results in a whitening of the PSD,
Qm Qm1 q
Sn
pω q,
as is
observed in both arctangent based discriminators, and has a significant impact on FLL performance. Taking a similar approach to the treatment of the discriminator mean, a Booton gain can be used to estimate the mean value of R1n . Due to the complexity of the discriminator function, examining (4.60), it is postulated that a closed form, similar to that of (4.58), is not currently tractable. Instead, a relatively simple approximating function has been found: R1n TL2
1 SNR
where the constants c1
c
erf pc1 SNRc q c2
c3 erf
a
0.5 SNRc
,
(4.66)
0.188, c2 0.694 and c3 0.306 have been found by a
MLS fit to the simulated data of Figure 4.12. It has also been stipulated in the choice 116
Section 4.3: Discriminator Analysis 0
Rn1 T2L
−0.05
−0.1
−0.15
Simulated Data −SNR−1 c
−0.2 −15
Approximate Model −10
−5
0
5 10 SNRc (dB)
15
20
25
30
Figure 4.12: The autocorrelation value, R1n , versus SNRc for a range of SNRc values for the decision directed cross product discriminator.
of these constants that c3
1 c2, such that the function converges to SNRc 1
for high SNRc values. A plot of this approximating function is shown in Figure 4.12 where it can be seen to fit well across the full range of SNRc . The ratio this approximate function, is plotted in Figure 4.9. It can be seen that
RR R1n
n 1 n 0
, using
reduces
in magnitude much more rapidly and at a higher value of SNRc for the decision directed cross product discriminator than for the cross product discriminator and that when SNRc
5 dB, the PSD, S npωq, is almost white for the decision directed
discriminator. As will be shown in Section 4.5, this effect results in poorer tracking
performance for FLLs employing a decision directed cross-product discriminator, when compared to those using a cross-product discriminator.
4.3.5
Approximations For The Arctangent Based Discriminators
As will be discussed in Section 4.5, the performance of the FLL for a given carrier frequency discriminator is defined by the three discriminator parameters: KD , R0n and R1n . In the case of the arctangent-based discriminators (Atan2 and Atan), expressions for these values have been presented, in Section 4.3, in the form of single, double and triple integrals, respectively. The integrals do not currently offer readily usable closed form solutions. Also, due to the nature of the discriminator functions, which exhibit step discontinuities (at at
π
and π for the Atan2 discriminator and
π, π{2, π{2 and π for the Atan discriminator), these integrals are relatively
difficult to evaluate numerically and, for high SNRc values, become quite oscillatory.
For these reasons, approximate expressions for these values prove useful in the design of FLLs. A set of approximate expressions for KD , R0n and R1n for the four-quadrant arctangent and the arctangent discriminators have been developed and are presented in Appendix A. 117
Chapter 4: Analysis and Design of Frequency Lock Loops
4.4
Impact of KD on Transient Response
In terms of transient response, the effect of the reduction in KD with reducing SNRc is evident in the step response of the system. Reducing KD causes the poles of (4.6) to move from their design positions, which they occupy at high SNRc values. This results in a change of Bω , β and η which can, in turn, result in a slower than expected loop response. For example, the simulated response of a critically damped, second order FLL with a two-sided bandwidth of Bω
2 Hz was examined for both high
(30 dB) and low (3.762 dB) SNRc conditions, using both the arctangent and decision directed cross product discriminators. The high SNRc case yields KD low SNRc case was chosen to result in
Atan KD
0.333 and
DDCP KD
1 while the
0.6, as can be
seen in Figures 4.4 and 4.11, respectively. The loop was subjected to a frequency step of 300 π rad/s and the transient response of the frequency estimate, ω ˆ , was recorded. The transient response of the arctangent loop is plotted in Figure 4.13 (a), wherein the curve labelled ‘High’ represents the high SNRc case and the curve labelled ‘Low’ represents the low SNRc case. It is evident that the response of the low SNRc case is slower and more oscillatory than that of the high SNRc case. This is, however, only one example of the transient response. It is relatively noisy and does not necessarily reflect the true impact of the KD reduction. More insight into the system can be gained by examining the mean step response. To find this, this frequency step was applied to the loop 200 times and the response was recorded. The two hundred recorded transient responses were then averaged. The mean step response is plotted in Figures 4.13 (b) and (c) for a loop using the arctangent and decision directed cross product discriminators, respectively. It can be seen that, as expected, the high SNRc cases (solid lines) for both discriminators are identical. The low SNRc cases, however, differ significantly. While both are slower and more oscillatory than the high SNRc cases, the effect is more significant in the arctangent discriminator loop. This is due to the reduction in KD for the arctangent discriminator being more severe (0.333 as opposed to 0.6). Atan Substituting KD
0.333 into (4.6), finding the system poles and calculating
the effective values of β and η, it is readily found that the decay-rate parameter, β, has reduced from a design value of 0.0016 to an effective value of 0.000535 and that the damping parameter, η, has changed from a critically damped value of η 2 an underdamped value of
η2
1.988.
0 to
The result is an effective Bω reduced from
2 Hz to 0.935 Hz. Likewise, for the decision directed cross product discriminator, β has reduced to a effective value of 0.000966 and η has changed to η 2 resulting in an effective Bω of 1.367 Hz.
0.655,
To compensate for the reduced discriminator gain, the loop filter gains, A0 and
A1 , should be increased by a factor of 1{KD , thereby restoring the system poles to their design values. An example of the KD -compensated loop response using 118
Section 4.5: Steady-State Performance of the FLL the arctangent discriminator is shown in Figure 4.13 (a) and labelled ‘Comp.’. The average transient responses were again calculated for both discriminators and are plotted in Figures 4.13 (b) and (c). It can be seen that the average transient response is almost restored to that of the high SNRc . The small difference in the response is due to the non-linearity of the discriminator functions. Under low SNRc conditions the linear regions of the discriminator functions reduce considerably, as discussed in Section 4.3, and this initial response to large frequency errors is diminished. It is also evident from Figure 4.13 (a) that the cost of this gain compensation is increased noise. The tracking error noise variance has increased in the KD compensated loop response, relative to the uncompensated low SNRc case. It is clear from this that the value of KD must be considered in FLL loop design. Failure to compensate for the noise-induced reduction in KD can result in a transient response which is far removed from that of the design point. By noting that KD appears in both the numerator and denominator of (4.6), it is clear that compensating for the reduction in KD by applying gains A0 {KD and A1 {KD to the loop filter will restore
the transfer function to its high SNRc form, as is evident from Figure 4.13. In con-
trast, KD appears only in the denominator of (4.5) and, therefore, compensating for KD results in an increase in tracking error variance. This cost is unavoidable and stems from the reduced signal-to-noise ratio of the carrier frequency estimate itself. A similar analysis can, of course, be applied to the four-quadrant arctangent discriminator and would show similar results, only differing in the value of KD at any given SNRc (as given by (4.44)). The cross product discriminator, in contrast, does not exhibit such transient response variation as the gain of the cross product discriminator is independent of the prevailing SNRc . Concise discussions of these effects can be found in [30, 31].
4.5
Steady-State Performance of the FLL
Using the expressions for discriminator gain and autocorrelation coefficients, developed in Section 4.3 and either (4.18) or (4.20), as appropriate, the steady state tracking error variance of the FLL can be estimated. Examining the tracking error variance of FLLs, using each of the four discriminators across a wide range of SNRc values, it is apparent that certain discriminators are better suited to particular SNRc conditions than others.
4.5.1
Tracking Error Variance
The tracking error variance of a 2 Hz, critically damped, second order FLL with a loop update period of TL
1 ms, using each of the four discriminators, was
estimated through Monte-Carlo simulation. Both simulated data (markers) and theoretical estimates (solid lines) are shown in Figure 4.14 for a range of SNRc 119
ω ˆ (rad/s)
Chapter 4: Analysis and Design of Frequency Lock Loops
1000 500 0 2
High 4
6
Low 8 Time (s)
Comp. 10
12
ω ˆ (rad/s)
(a) Single response using the arctangent discriminator
1000 500 0 2
High 4
6
Low 8 Time (s)
Comp. 10
12
ω ˆ (rad/s)
(b) Mean response using the arctangent discriminator
1000 500 0 2
High 4
6
Low 8 Time (s)
Comp. 10
12
(c) Mean response using the decision directed cross product discriminator Figure 4.13: Response of FLL to frequency step of 300π rad/s estimated through Monte-Carlo simulation.
120
Section 4.5: Steady-State Performance of the FLL 6
10
ATAN2 CP ATAN DDCP
4
σ2ω (rad2/s2)
10
2
10
ATAN vs DDCP Threshold SNRc ≈ 10 dB
ATAN2 vs CP Threshold SNRc ≈ 6 dB
0
10
−5
0
5
10 SNRc (dB)
15
20
Figure 4.14: The tracking error variance for a second order, 2 Hz bandwidth, critically damped FLL using a loop update period of 1 ms using each of the four discriminators.
values. The simulated and theoretical results agree well and illustrate that the tracking error variance is not a linear function of SNRc . It appears that the tracking error variance of the two arctangent based loops converge for high SNRc values. This is to be expected as the autocorrelation values of these discriminators converge for high SNRc . A similar convergence is observed in the cross product discriminators, although it occurs at a different SNRc value. The relationship between tracking error variance and SNRc is not linear for any of the four discriminators. The arctangent discriminators exhibit an inversely proportional relationship to SNRc when SNRc is greater than approximately 12 dB and 15 dB for the four-quadrant arctangent and arctangent discriminators, respectively. Below this SNRc value, the tracking error variance increases significantly. For the cross product discriminator, the increase in tracking error variance is smoother and does not exhibit any sudden changes of slope. The relative tracking error variance incurred by FLLs employing each of these discriminators is further discussed in [30, 31].
4.5.2
Comparisons of Steady-State Performance
Of particular interest is the ratio between the tracking error variance of each discriminator at a given SNRc . Of course, all four discriminators cannot be compared fairly to one another, but, rather, the coherent discriminators (CP and ATAN2) can be compared and the non-coherent discriminators (ATAN and DDCP) can be compared. From Figure 4.14, it is clear that the arctangent discriminators achieve 121
Chapter 4: Analysis and Design of Frequency Lock Loops a significantly lower tracking error variance when SNRc is high. As SNRc is reduced and the tracking error variance of the arctangent based discriminators undergoes a marked increase, the variance exceeds that of the corresponding cross product discriminators. Plots of the ratio of the four-quadrant arctangent discriminator tracking error variance to that of the cross product discriminator and of the arctangent discriminator to that of the decision directed cross product discriminator are shown in Figures 4.15 (a) and (b), respectively. These curves relate to the FLL configuration of Figure 4.14 (2 Hz) and to two similar FLL loop configurations with 10 Hz and 20 Hz bandwidths, respectively. The difference in tracking error variance is significant. The four-quadrant arctangent discriminator can outperform the crossproduct discriminator by more than 10 dB in high SNRc environments and, for low SNRc environments, the cross-product discriminator can offer a 5 dB improvement over four-quadrant arctangent discriminator. A similar trend is observed for the arctangent and decision directed cross-product discriminators. Clearly, then, it is important to consider the prevailing SNRc when choosing a discriminator function. The SNRc thresholds at which the cross product discriminators outperform the corresponding arctangent discriminators are annotated in Figure 4.14. These points correspond to the zero crossings of the curves in Figure 4.15 and are approximately 6 dB and 10 dB for the coherent and the non-coherent discriminators, respectively. In general, the thresholds can be found using (4.18) or (4.20) and solving for SNRc such that:
σω2 AT AN 2 ,Rn RnAT AN 2 KD KD σω2 AT AN ,Rn RnAT AN KD KD
σω2 CP ,Rn RnCP KD KD
(4.67)
σω2 . DDCP ,Rn RnDDCP KD KD
(4.68)
As can be seen from (4.18) and (4.20), the threshold is a function not only of KD and Rn , but also of the loop filter gain, A0 and, in the second order case, A1 . The threshold will, therefore, change for different loop bandwidths. Considering the form of the gains and autocorrelation functions of the various discriminators, a solution in terms of the filter gains currently appears to be intractable and unlikely to be informative. Rather, the threshold can be expressed in terms of the normalised FLL bandwidth, Bω TL , for three common filter configurations. Figure 4.16 shows the threshold for the coherent and non-coherent discriminators for a first order FLL and two second order FLLs (critically damped and standard underdamped), for a range of normalised FLL bandwidths. When the prevailing SNRc is above the threshold, the arctangent based discriminators should be used, otherwise, the cross product based discriminators should be used. As can be seen in Figure 4.16, for the non-coherent discriminators the thresh122
Section 4.5: Steady-State Performance of the FLL
Variace Ratio (dB)
5
0
−5 2 Hz 10 Hz 20 Hz
−10
−15
−10
−5
0
5 10 SNRc (dB)
15
20
25
30
(a) Ratio of σω2 for FLLs using the ATAN2 and CP discriminators ( ATAN2 ). CP
Variace Ratio (dB)
5
0
−5 2 Hz 10 Hz 20 Hz
−10
−15
−10
−5
0
5 10 SNRc (dB)
15
20
25
30
ATAN ). (b) Ratio of σω2 for FLLs using the ATAN and DDCP discriminators ( DDCP
Figure 4.15: The ratio of tracking error variance in decibels for (a) the ATAN2 and CP discriminators and (b) the ATAN and DDCP discriminators for each of three filter bandwidths, 2 Hz, 10 Hz and 20 Hz.
123
Chapter 4: Analysis and Design of Frequency Lock Loops
11
SNRc Threshold (dB)
10
ATAN2 | CP
9 8 7
1st Order 2nd Order, η2 = 0
6
2nd Order, η2 = −1
ATAN | DDCP 5
0
0.2
0.4
0.6
0.8
1
Bω TL Figure 4.16: The threshold at which the the arctangent and cross product discriminators yield equal tracking error variance plotted against the normalised bandwidth, Bω TL , for FLLs using first order, second order critically damped, and second order standard underdamped loop filters.
old does not vary significantly with Bω TL or with different loop filters, remaining approximately equal to 10.5 dB. It is reasonable, therefore, to use a hard limit of, say, 10.5 dB for the coherent threshold, regardless of loop design. For the coherent discriminators, however, the threshold changes considerably across the different loop filters and with changing Bω TL , in particular in the region 0
Bω TL ¤ 0.3,
which is most commonly the case for GNSS receivers [61, 80]. As the curves in Figure 4.15 (a) are quite steep in the vicinity of the zero crossing, the threshold, specific to the loop design and bandwidth, must be used to fully exploit the discriminator characteristics.
4.6
A note on SNRc
This work has considered the design and performance of the FLL when applied to the ideal GNSS signal, in the absence of front-end filtering and quantisation effects. Of course, results here are applicable to the real GNSS receiver, which incur significant processing losses. It has been shown in Chapter 3 that the fontend filter and the quantiser effect the shape of the cross-correlation between the satellite signals spreading code and the local replica code. That is, they effect the
relationship between y rk s and δτ . The relationship between y rk s and both δθ and
δω remains unchanged. Results presented in this chapter relate FLL performance, in the presence of thermal noise, to one property of y rk s, namely SNRc . Thus,
the performance of the FLL for a real receiver can be estimated, using the theory presented here, by calculating SNRc using the appropriate expressions presented in 124
Section 4.7: Conclusions Chapter 3. When designing FLL loop filters, a designer must often choose a loop design based on an estimate of what the received signal power will be (e.g. MRSP for the GPS L1 C/A signal is -158.5 dBW). To employ the design rules presented here, the
predicted value of SNRc is given by L SNRIdeal , where L is calculated using the c
appropriate expressions in Chapter 3. For example, assuming a thermal noise floor is
equal to -204 dB / Hz, a MRSP of -158.5 dBW, the receiver front-end configuration of Table 3.1 and-one bit quantisation, it can be shown that: SNRIdeal c
L 1.18 dB and SNRc
17.32 dB.
18.5 dB,
In other cases, the receiver will automatically produce a live estimate of the signal strength, either from the acquisition stage or from a previous tracking stage. In such a case, an estimate of SNRc is made directly from previous samples of y rk s which, of course, will inherently account for any preceding receiver losses, and can be used to choose suitable FLL design parameters.
4.7
Conclusions
This chapter has considered the design and performance of FLLs for GNSS receivers. Following a thorough analysis of the FLL, it has been shown that certain key design choices and certain noise induced effects can have a significant impact on the overall FLL performance. The design of loop filters has been addressed and novel expressions for filter coefficients for a first order, and two popular second order loop filters, have been presented in Section 4.2. The properties of four carrier frequency discriminators have been examined in Section 4.3 and, based on these results, the transient performance of the FLL in the presence of thermal noise has been studied in Section 4.4. It is clear that, under weak signal conditions, the performance of the FLL can be severely degraded and that the transient performance can differ significantly from that of the design point. A design procedure mitigating this effect is presented and verified. Using new closed form expressions for the tracking error variance of the FLL in the presence of thermal noise, presented in Section 4.1.2, and the discriminator analysis, presented in Section 4.3, a comparison of the four different discriminators has been presented. As shown in Section 4.5.1, the relationship between signal-to-noise ratio and tracking error variance is nonlinear and is unique to each discriminator, as dictated by the three discriminator parameters: KD , R0n and R1n . Utilizing this analysis it is shown in Section 4.5.2 that, by choosing a discriminator function based on the prevailing signal-to-noise ratio (SNRc ), a significant performance improvement can be achieved. The improvements can be quite significant, exceeding 10 dB in certain cases. The threshold at which the discriminator should be changed has been found to be approximately related to the normalised FLL bandwidth and that, in the case of the non-coherent discriminators, it is relatively constant across the 125
Chapter 4: Analysis and Design of Frequency Lock Loops range of interest. The treatment of the FLL and frequency discriminators used here does not make many assumptions about the discriminator functions but, rather, hinges the analysis on the three parameters KD , R0n and R1n . Provided that the discriminator produces frequency estimates based on the difference between successive phase estimates, and that the linear region of the discriminator is reasonably large (i.e. comparable to one quarter of the reciprocal of the loop update period), then this analysis can be extended to arbitrary discriminator functions. Once KD , R0n and R1n have been found, a range of FLLs could be compared using the same metrics. Moreover, the performance of the loops could be further enhanced, should new discriminators prove superior for certain SNRc regions. Frequency is often the first property of the carrier to be estimated by the receiver. Initially (and simultaneous to the estimation of code phase) the carrier frequency is coarsely estimated by the acquisition algorithm. Using the mechanism examined in this chapter, it is refined though recursive processing in the FLL. While the FLL can offer reliable and robust estimates of the carrier frequency, it is often insufficient. Neither is it the only carrier parameter that can be estimated; it is also possible to produce direct estimates of the carrier phase. Carrier phase estimation can provide more accurate frequency estimates, facilitate more reliable data demodulation and provide more accurate PVT estimates than frequency estimation alone. Hence, the FLL is commonly employed as an intermediate step between signal acquisition and carrier phase tracking. It is appropriate, therefore, having studied the FLL in this chapter, to proceed to an examination of the PLL in the next chapter.
126
Chapter 5
The Phase Lock Loop I: Design and Performance Analysis Chapter 4 has considered the problem of carrier frequency tracking in the presence of thermal noise, using the FLL. While this approach to carrier synchronisation is useful and quite robust, as discussed in Chapter 2, carrier phase tracking has some advantages over carrier frequency tracking. For example, the coherent data demodulation techniques can be applied, as opposed to non-coherent techniques. Also, in terms of PVT measurements, more accurate Doppler estimates can be extracted and carrier phase positioning techniques can be applied. This chapter presents a thorough analysis of carrier phase estimation in the presence of thermal noise interference. Similar to the discriminator model presented in Chapter 4, it is shown that the carrier phase discriminator can be characterised in terms of three metrics: gain, variance and linear region. This discriminator model is utilized in an analysis of a PLL operating in its linear region and the implications of the discriminator on the closed loop performance are examined. It is shown that the choice of discriminator which yields optimum performance is dependent on the prevailing signal-to-noise ratio. The optimum discriminator choice is presented for each of a number of noteworthy operating regions. Following this linear analysis, the process of PLL acquisition, which exercises the non-linear region of operation of the discriminator, is considered. The process of phase estimation and tracking from the instant of handover from the FLL to the PLL is examined through Monte-Carlo simulation. The performance of the four carrier phase discriminators in the acquisition stage is examined in terms of the time taken for the PLL to settle. It is shown that, within the range of operating conditions considered in this experiment, the choice of discriminator, which minimises the time taken for the PLL to settle, is a function of the prevailing carrier-to-noise ratio and of the loop bandwidth. Results indicate that this preferred discriminator is not, necessarily, the preferred choice of discriminator for linear operation. It is demonstrated that the appropriate choice of discriminator can yield a significant 127
Chapter 5: The Phase Lock Loop I reduction in settling time.
5.1
Receiver Model and PLL Architecture
A full analysis of the PLL in the presence of receiver non-idealities, signal propagation effects, user dynamics and RF interference is, generally, not mathematically tractable and is not attempted here. Rather, an analysis of the PLL under the influence of the dominant interference is presented. In this chapter, thermal noise and simple, deterministic, dynamics are considered. Stochastic signal phase dynamics, such as those induced by the local oscillator and by ionospheric and tropospheric scintillation, are neglected at present and will be considered in Chapter 6. The effects of multi-path propagation are also neglected. This section introduces a simplified model of the received signal and the corresponding correlator values, utilizing the analysis presented in Chapter 3. A generalised, high-level, description of the classical PLL is introduced, based on which a linearised system describing the PLL operation is assembled. This model forms the subject of the PLL analysis presented in the first half of the chapter.
5.1.1
Down-Conversion and IF Signal Processing
Similar to Chapters 3 and 4, the digital, sampled and quantised signal at IF can be expressed as:
rIF rk s sIF rk s
sIF rk s
?
n rk s
2P cos pωc kTs
(5.1) θ rk s Ψ rk sq c pkTs τ q d pkTs τ q .
(5.2)
The sequence θrk s represents the signal phase dynamics, including satellite induced
Doppler, Doppler drift and user-dynamic induced phase variations. Oscillator in-
duced phase dynamics, neglected in the previous chapters, are considered here, rep-
resented by Ψrk s and are discussed further in Chapter 6. Again, in a similar fashion
to the earlier chapters, the correlation of the local replica signals with the incoming signal can be approximated by:
Im Qm
dm
?
δωc TL Rcode pδτ q cos δθ 2
? δωc TL dm P sinc 2 Rcode pδτ q sin δθ P sinc
δωc TL 2
δωc TL 2
ni rms
(5.3)
nq rms ,
(5.4)
where δω and δθ are redefined slightly, compared to the previous chapters, to denote the mean code phase, carrier frequency and carrier phase errors, respectively, including the phase and frequency errors induced by the Ψ rk s term. The correlator noise 128
Section 5.1: Receiver Model and PLL Architecture
Sum and Dump
Σ
Im Discriminator
r[k]
KD
Σ
c[k]
Qm
From Code Tracking Loop
cos
OSC
e Loop Filter
sin
NCO
Figure 5.1: Block Diagram of a Standard PLL
process, however, is unaffected by the oscillator phase instability, and is defined as in Chapter 3. An estimate of the carrier phase tracking error, δθn , is then made by applying a carrier phase discriminator to the values Im and Qm . This estimation procedure is discussed in more detail in Section 5.2.
5.1.2
The Phase-Lock Loop
The standard phase-lock loop is a feedback control loop which tracks the carrier phase using estimates of the carrier phase tracking error. A typical PLL consists of a pair of correlators, to produce Im and Qm , a carrier phase discriminator, a loop filter and a numerically controlled oscillator. A block diagram of a typical PLL structure is shown in Figure 5.1. As discussed in Section 5.2, the phase tracking error estimator is nonlinear. However, if this estimate is linearised around zero phase error, and normalised such that the noise-free estimate has unity gain, the phase error estimate, denoted em , can be expressed as: em
KD δθm
nθm .
(5.5)
That is, the phase error estimate is approximately equal to a constant times the actual phase error, plus a zero mean, white noise, nθ . The constant gain, KD , is referred to as the discriminator gain and depends on the chosen discriminator function and the prevailing signal-to-noise-ratio. The variance of nθ is also dependent on the phase discriminator used and the received signal-to-noise ratio. It has a minimum variance equal to the variance of nq (this occurs when Qm is used as the phase error discriminator). The two-sided spectral density of nθ is denoted here by Nθ , which, assuming correct normalisation of the discriminator gain and 129
Chapter 5: The Phase Lock Loop I a high signal-to-noise ratio, is approximately equal to half of the reciprocal of the carrier-to-noise-floor-ratio. The exact details of the linearisation of this phase error estimate and the values of the PSD of nθ for various discriminators will be given in Section 5.2. The remainder of the PLL is linear and can be represented by a system of zdomain transfer functions, where the update interval of the system is TL . Such a linearised loop model is useful as it facilitates the estimation of loop stability and tracking performance. Of particular interest are the transfer functions between the thermal noise, nθ , and the tracking error, δθ, between the carrier phase, θ, or oscillator phase noise, Ψ, and the tracking error, δθ, and between the carrier phase ˆ These quantities are or oscillator phase noise and the carrier phase estimate, θ. depicted in a linearised loop model in Figure 5.2. The three transfer functions of interest are given by:
Hn pz q Hθ pz q Hθˆpz q
∆Θpz q N COpzqF pzq Nθ pz q 1 KD N COpz qF pz q ∆Θpz q ∆Θpz q 1 Θpz q Ψpz q 1 KD N COpz qF pz q ˆ pz q ˆ pz q Θ Θ KD N COpz qF pz q , Θpz q Ψpz q 1 KD N COpz qF pz q
(5.6) (5.7) (5.8)
where upper-case symbols represent the z-transform of the corresponding lower-
case time series. The functions F pz q and N COpz q represent the z-transform of the
loop filter and the numerically controlled oscillator, respectively. The numerically controlled oscillator, essentially a digital integrator and a gain, is defined as: N COpz q
zTL . z1
(5.9)
Figure 5.3 depicts a generalized PI controller, as is typically used in GNSS PLLs. The filter takes the form: F pz q
P ¸
Ak
p 0
Ts z z1
p
(5.10)
.
Such a controller facilitates direct measurement of signal dynamics; the signal at point ‘a’ will provide an estimate of carrier frequency (Doppler) and the signal at point ‘b’ will provide an estimate of the rate of change of carrier frequency (Doppler drift). Although higher order loop filters are possible (as indicated by the dotted lines), typically a second order loop (P sufficient.
130
1) or a third order loop (P
2) is
Section 5.1: Receiver Model and PLL Architecture
nθ
θ +ψ
δθ
+
+ +
KD
e F(z)
-
θˆ NCO(z) Figure 5.2: Linearised PLL Model
x[m]
+
A0
y[m]
+ A1
A2
TL ∑ −k ∞
a
TL ∑ −k ∞
+ +
TL ∑ −k ∞
b +
Figure 5.3: A generalized PI controller
131
+
Chapter 5: The Phase Lock Loop I
5.2
Carrier Phase Estimation
As discussed in Section 5.1.2, the performance of the PLL in the presence of AWGN can be estimated by examining the linear model and the noise performance of the carrier phase discriminator (similar to the treatment of the FLL in Chapter 4). Four popular carrier phase discriminators are examined here: the four-quadrant arctangent discriminator, the arctangent discriminator, the decision-directed discriminator (often referred to as the Costas discriminator) and the simple quadrature discriminator. The discriminators are characterised in terms of gain, KD and variance, σn2 θ , (in Section 5.2.5 the linear region will also be considered) where: B E res KD Bδθ δθ0 2 σn Var rnθ s Var res θ
(5.11)
(5.12)
.
δθ 0
In the carrier phase discriminator analysis that follows, it is assumed that the PLL is operating correctly such that the mean frequency error is zero, and the maximum frequency error is reasonably small, relative to the update interval (such that the phase error accrued over the update interval is small, less than the linear region of the discriminator, for example). It is also assumed that the code tracking error is zero and that the effects of quantisation and filtering can be embodied in a loss coefficient applied to the received signal-to-noise ratio, as detailed in Chapter 3. Under these circumstances sinc
δωc TL 2
Rcode pδτ q
1. Also, if the phase error
induced by the mean frequency offset over the correlation interval,
δωc TL 2 ,
is absorbed
into the mean phase error, δθ then the correlator values, given in (5.4), simplify to:
? dm P cos pδθq ? Qm dm P sin pδθq Im
ni rms
nq rms .
(5.13) (5.14)
These expressions will now be employed in the characterisation of the four carrier phase discriminators.
5.2.1
The four-quadrant arctangent discriminator
The four-quadrant arctangent discriminator is defined as: eAtan2 m
arctan pIm, Qmq .
(5.15)
The mean response of this discriminator to phase error, denoted here by µe , is shown in Figure 5.4, for a variety of SNRc values. It is evident that, for high values of SNRc , µe is relatively linear across a wide range of phase error values 132
Section 5.2: Carrier Phase Estimation
3 2
µe (rad)
1 0 Noise Free 8 dB 3 dB −2 dB −7 dB
−1 −2 −3 −3
−2
−1
0 δθ (rad)
1
2
3
Figure 5.4: Mean response of the four-quadrant arctangent discriminator versus true input phase error for a selection of SNRc values.
and has approximately unity gain. As the value of SNRc reduces, the gain reduces considerably and the linear region diminishes. Examining (5.15), it can be seen that the phase error estimate is a function of only the current correlator values (Im and Qm ). As Im and Qm are corrupted by white noise, the PSD of eAtan2 will also be m white. This fact, it will be shown here, greatly simplifies the analysis of the noise performance of the PLL as compared to the FLL. The value of µe , as depicted in Figure 5.4, can be calculated using the probability
density function ppφq given in (4.41) and re-produced here for convenience: c SNRc 1 e 2 π SNRc 2 ppφq 1 cospφqe 2 SNRc cospφq 2π 2
SNRc cospφq ? 1 erf , π ¤ φ ¤ π. 2
(5.16)
Similar to the approach taken when Chapter 4 in evaluating the response of the four-quadrant arctangent frequency discriminator, it is readily shown that:
µAtan2 e
»π
π »π
π
arctan pcos pδθ
φq , sin pδθ
φqq p pφq dφ
φp pφ δθq dφ.
From (5.11), taking the first derivative of (5.17) and setting δθ 133
(5.17)
0, the discrim-
Chapter 5: The Phase Lock Loop I
1
KD
0.8 0.6 0.4 Atan2 Atan Sign(I).Q Q
0.2 0 −15
−10
−5
0
5 10 SNRc (dB)
15
20
25
30
Figure 5.5: Discriminator gain (KD ) versus SNRc for the four-quadrant arctangent discriminator (Atan2), the arctangent discriminator (Atan), decision-directed discriminator (Sgn(I).Q) and the quadrature discriminator (Q).
inator gain is found to be:
Atan2 KD
B µAtan2 e Bδθ δθ0
(5.18)
φp1 pφq dφ,
(5.19)
»π
π
where p1 pφq is the first derivative of p pφq w.r.t. φ and is given by:
p1 pφq
SNRc e 2 a SNRc sinpφq 4π
erf
?
1
?SNR cospφq
?c 2
2
2πe 2 SNRc cos
1
a
pφq SNRc cos2 pφq
1
2 SNRc cospφq .
(5.20)
A plot of KD versus SNRc for this discriminator is shown in Figure 5.5. It is evident that for SNRc values below approximately 6 dB, the discriminator gain reduces rapidly. This reduction in discriminator gain has implications for the closed loop poles of (5.8) and (5.6), similar to the case of the FLL, as discussed in Chapter 4. When operating a PLL employing the four-quadrant when the prevailing SNRc is below 6 dB the gain of the loop filter coefficients must be scaled by the reciprocal of KD (as given by (5.19)) to sustain the desired closed loop performance. In a similar fashion to the mean of the four-quadrant arctangent discriminator, 134
Section 5.2: Carrier Phase Estimation
1
10
1/SNRc Atan2 Atan
Var[nθ] (rad2)
0
10
−1
10
π2 3
π2 12
−2
10
−3
10 −20
−10
0
10 SNRc (dB)
20
30
Figure 5.6: Variance of the carrier phase estimate versus SNRc , linearised around a zero phase error, for the four-quadrant arctangent discriminator and the arctangent discriminator.
the variance of this carrier phase estimate can be found via:
Var n
θ
»π
π
arctan pcos pδθ
φq , sin pδθ
φqq2 p pφq dφ.
(5.21)
This variance estimate, however, is a function of δθ. Assuming that the PLL is operating correctly, and that the loop is tracking with an approximately zero mean phase error, it is useful to linearise this estimate around δθ
Var n
θ
»π
π »π
π
0:
arctan pcos pφq , sin pφqq2 p pφq dφ φ2 p pφq dφ.
(5.22)
A plot of Var nθ for this discriminator is shown in Figure 5.6. Unsurprisingly, the discriminator variance changes linearly with SNRc for high SNRc values, bearing
the approximate relationship: Var nθ imately Gaussian. At SNRc
1{ SNRc.
In this region (5.16) is approx-
11 dB the discriminator variance begins to increase
beyond 1{ SNRc with reducing SNRc . At this point, (5.16) has begun to resemble a truncated Gaussian distribution. As SNRc is further reduced, (5.16) approaches
a uniform distribution over the interval of
π2
{3.
rπ, πs and reaches a maximum variance
This nonlinear relationship between SNRc and Var nθ has a significant 135
Chapter 5: The Phase Lock Loop I
µe (rad)
1
0
−1
Noise Free
−2 −3
−2
12 dB −1
8 dB
0 δθ (rad)
4 dB 1
2
0 dB 3
Figure 5.7: Mean response of the arctangent discriminator versus true input phase error for a selection of SNRc values.
impact on the performance of the PLL under weak signal conditions and, in conjunction with the discriminator gain effects described earlier, can result in severely degraded tracking performance. These effects must, therefore, be considered in the design of the PLL and will be discussed further in Section 5.3.
5.2.2
The arctangent discriminator
The arctangent discriminator is defined as: eAtan m
arctan pQm{Imq .
(5.23)
The mean response of this discriminator to δθ, again denoted here by µe , is shown in Figure 5.7, for a range of SNRc values. Similar to the four-quadrant arctangent discriminator, for high values of SNRc , µe changes in a relatively linear fashion with changing δθ and has approximately unity gain. As the value of SNRc reduces, the gain reduces considerably and the linear region diminishes. This occurs at a higher SNRc value for the arctangent discriminator than for the four-quadrant arctangent discriminator, owing to its smaller linear region (discussed further in Section 5.2.5). Similar to Section 5.2.1, it can be shown that the mean response of the arctangent discriminator, after some simplification, is given by: 136
Section 5.2: Carrier Phase Estimation
µAtan e
»π
arctan
π »π
sin pδθ cos pδθ
φq φq
φ pp pφ δθq p pφ
p pφq dφ
δθqq dφ
0
»π
π pp pφ δθq p pφ
δθqq dφ,
(5.24)
π 2
where the limits of integration have been manipulated such that the arctangent function and its arguments reduce to simple linear combinations of δθ, φ and π. Again, from (5.11), taking the first derivative of (5.24) and setting δθ arctangent discriminator gain,
Atan KD
Atan , KD
is found to be:
B µAtan e Bδθ δθ0
2
»π
0, the
(5.25)
φp1 pφqdφ π
»π
p1 pφqdφ .
(5.26)
π 2
0
Figure 5.5 depicts the relationship between KD and SNRc for this discriminator. For SNRc values below approximately 10 dB, the discriminator gain reduces rapidly. Although the trend is similar to that of the four-quadrant arctangent discriminator, it occurs at a higher SNRc value and the reduction in KD with SNRc is steeper. As is the case for the four-quadrant arctangent discriminator, and indeed for all discriminators which exhibit a variation of gain with SNRc , some gain compensation must be applied to the discriminator estimates to maintain the high-SNRc transient performance. Similar to the four-quadrant arctangent discriminator, the variance of this carrier phase estimate, linearised around a zero phase error, can be found via:
Var nθ
»π 2 φ2 p φ dφ
pq
»π
π
pπ 2φqppφq dφ .
(5.27)
π 2
0
Figure 5.6 illustrates this relationship across an appropriate range of SNRc values. Again, similar to the four-quadrant arctangent discriminator, the discriminator variance changes linearly with SNRc for high SNRc values. As SNRc is reduced, (5.16) approaches a uniform distribution over the interval r π2 , variance of
π2 12 .
137
π 2
s and reaches a maximum
Chapter 5: The Phase Lock Loop I
µe (rad)
1
0
−1 Noise Free −3
−2
12 dB −1
8 dB
0 δθ (rad)
4 dB 1
2
0 dB 3
Figure 5.8: Mean response of the decision-directed discriminator versus true input phase error for a selection of SNRc values.
5.2.3
The decision-directed (Costas) discriminator
The decision-directed discriminator is defined as:
pI q.Q eSign m
sgn pImq Qm.
(5.28)
The function of the sgn pIm q term in this discriminator function is to render it
insensitive to data modulation. The Qm term provides an estimate of δθ multiplied
by the data value dm while the sgn pIm q term provides an estimate of the dm . As d2m
1, this discriminator is (ideally) insensitive to data modulation. A plot of the
p q , denoted µSignpI q.Q , is shown in Figure 5.8. The value of e
Sign I .Q
mean value of em
p q can be found from:
Sign I .Q
µe
pI q.Q µSign e
E rsgn pImq Qms .
(5.29)
Recalling from Chapter 3 that < ty rmsu and = ty rmsu are statistically indepen-
dent, then:
pI q.Q µSign e
E rsgn pImqs E rQms ? E rsgn pImqs dm P sin pδθq .
(5.30)
The E rsgn pIm qs component can be evaluated in a similar manner to the one-bit quantisation case of Section 3.4.1. The problem is reduced to the estimation of dm given the AWGN corrupted sample: shown that:
?
P cos pδθq dm
138
ni . It can, thus, be readily
Section 5.2: Carrier Phase Estimation
pI q.Q µSign e
?
c
P erf
SNRc cos pδθq sin pδθq . 2
(5.31)
Unlike the arctangent-based discriminator functions, this phase estimate is not selfnormalising, that is, the estimate is a function of the nominal received signal power. To use this discriminator, even for high SNRc values (where the arctangent-based discriminators are completely self normalising), this phase estimate must be normalised by an estimate of The gain,
pq
Sign I .Q KD ,
?
P.
of this discriminator can be shown, using (5.11) and (5.31),
to be:
pq
Sign I .Q KD
c
?
P erf
SNRc 2
,
where, as before, the gain is linearised around an operating point of δθ of KD versus SNRc is shown in Figure 5.5.
(5.32)
0. A plot
Unlike the arctangent based discriminators, the variance of the decision-directed discriminator can be readily related to SNRc . From (5.12) and (5.28), we find:
Var nθ
E sgn pImq2 Q2mδθ0 E Q2mδθ0 Var rQms P SNR .
(5.33)
c
Of course, under relatively high SNRc conditions, and when the carrier phase
?
estimate has been correctly normalised by 1{ P , then the variance of this carrier
phase estimate is given by 1{ SNRc . This is illustrated in Figure 5.6, providing a comparison with the arctangent based discriminators.
5.2.4
The quadrature discriminator
The quadrature discriminator is defined, as its name suggests, as: eQ m
Qm.
(5.34)
This discriminator function is, by far, the simplest form of carrier phase estimator. Similar to the four-quadrant arctangent discriminator, it is sensitive to data modulation. Owing to its simple definition, the characteristics of this discriminator are quite easily expressed, having a mean value, µQ e , of: 139
Chapter 5: The Phase Lock Loop I
µQ e
E rQms ? dm P sin pδθq .
(5.35)
Q The discriminator gain, KD , is given by:
Q KD
?
(5.36)
P,
which, unlike the previous three discriminators, is independent of SNRc . The variance of this carrier phase estimate can be found in a similar manner to (5.33), and is given by:
Var nθ
E Q2mδθ0 Var rQms P . SNR
(5.37)
c
Again, this can be compared to the variance of the arctangent discriminators in Figure 5.6.
5.2.5
The discriminator Linear Region
The analysis of carrier phase estimation in the previous sections has considered a linearised model of the discriminator. In particular, the discriminator gain has been linearised about a zero phase error. While knowledge of the discriminator gain proves useful when predicting closed loop performance, more information is needed to adequately characterise the discriminator. A linearisation of the discriminator phase estimate around a zero phase error, given by µe
KD δθ, is accurate only
for a limited range of δθ. The region over which this approximation is acceptable is termed the linear region. All discriminators have a finite linear region, indeed, the linear region is ultimately limited to the range
rπ, πs, owing to the periodic
nature of the sinusoid. As has been shown in the previous sections, the values of KD and (µe ) are dependent on the discriminator function, and, with the exception of the quadrature discriminator, are also dependent on SNRc . It is unsurprising, therefore, that the linear region should also depend on the discriminator function and SNRc . In general, the linear region is symmetric around the origin and so it can be
defined by the single scalar LR such that the linear region is the interval rLR, LRs.
LR is defined as the value of δθ at which the true value of µe and the approximation 140
Section 5.2: Carrier Phase Estimation
2 1.8
µAtan2 e
1.6
KDδθ
1.4
0.9KDδθ
1.2 1 0.8 0.6
LR10% ≈ 1.7
0.4 0.2 0 0
0.5
1
1.5 δθ (rad)
2
2.5
3
Figure 5.9: Illustration of the 10% linear region for the four-quadrant arctangent discriminator for SNRc 3 dB.
µe
KD δθ differ by a certain percentage. The percentage is chosen arbitrarily, often
depending on the application, but, typical values are 1%, 5% and 10%. Specifically, LR for an x% linear region, denoted LRx% , is defined as(1) : LRx%
"
KD δθ δθ P R : µe
1
x , δθ 100
*
¡0
.
(5.38)
A graphical illustration of the solution to (5.38) is shown in Figure 5.9. This example considers the four-quadrant arctangent discriminator for SNRc this condition KD
3 dB. Under
0.9 yielding the linear approximation (µe KD δθ) represented
by the green broken line. The 10% bound is given by 0.9KD δθ and represented by the broken red line. It is, thus, the intersection of the true discriminator curve and the line 0.9KD δθ which defines the linear region. In this particular example, LR10%
1.7.
A plot of LR1% , LR5% and LR10% versus SNRc is shown in Figure 5.10. It is evident that the LR trend with changing SNRc is similar in each case, differing, primarily, in magnitude. Examining the coherent discriminators, it can be seen that the four-quadrant arctangent discriminator has a significantly larger linear region than the quadrature discriminator over the entire SNRc range of interest. For the non-coherent case, the arctangent discriminator has a significantly larger linear region than the decision-directed discriminator, for high SNRc values. For SNRc (1) The notation A tB P C : Du can be interpreted as A equals values of B in the set C such that condition D is satisfied.
141
Chapter 5: The Phase Lock Loop I values below approximately 7 dB, however, the linear region of both discriminators converge. The implications of the specific LR values and their dependence on SNRc will be discussed further in Section 5.3.2 and 5.4.
5.3
Closed Loop Performance: Linear Operation
The implications of the carrier phase discriminator’s gain and variance on the closed loop performance of the PLL is considered in this section. Given the stark differences that exist between the characteristics of the four discriminators presented, it is not unsurprising that some perform better than others in various different operating scenarios. The problem of choosing a discriminator based on the prevailing conditions is examined here. To perform a comparative analysis of the discriminators, first a performance metric must be defined. Of course, if a performance metric can be defined in one variable, then the comparison is simple. Therefore, the definition used in this section is as follows: if two PLLs, using different carrier phase discriminators, are tuned to provide the same mean dynamic response, then the one which yields lower thermal noise induced steady state tracking error is the better discriminator choice. Put simply, if two PLLs are capable of the same dynamic performance but one exhibits better noise performance, then it is the better PLL. This performance metric is, however, rather restricted, in that it assumes that the PLL must always remain in its linear region. While this condition is satisfied for the majority of the time, there are some occasions when the PLL operates outside this region. These occasions are examined in Section 5.4. The dynamic response of the PLL is considered in terms of its phase step response, its frequency step response and, subsequently, its equivalent noise bandwidth. For simplicity, the second order PLL is considered, due, in part, to its popularity in current GNSS receiver technology and in part to the tractability of its mathematical representation.
5.3.1
Impact of KD on Transient Response
Given the expressions for KD derived in Section 5.2, this section examines the impact of the discriminator gain on the transient performance of the PLL when it is operating in its linear region. To operate in its linear region, the maximum absolute phase error must not exceed LR during the phase transient. When this condition is satisfied, the transient response of the PLL is well approximated by traditional linear system theory. The response of the second order PLL to step changes in θ and ωc is examined here. The relationship between the received phase and δθ is described by the transfer function (5.7), repeated here for convenience: 142
Section 5.3: Closed Loop Performance: Linear Operation
LR10% (rad)
3 2
Atan2 Atan Sign(I).Q Q
1 0 −10
LR5% (rad)
3 2
0
10 SNRc (dB)
20
30
0
10 SNRc (dB)
20
30
0
10 SNRc (dB)
20
30
Atan2 Atan Sign(I).Q Q
1 0 −10
LR1% (rad)
3 2 1
Atan2 Atan Sign(I).Q Q
0 −10
Figure 5.10: LR1% , LR5% and LR10% versus SNRc for the four-quadrant arctangent discriminator (Atan2), the arctangent discriminator (Atan), decision-directed discriminator (Sign(I).Q) and the quardature discriminator (Q).
143
Chapter 5: The Phase Lock Loop I
∆Θpz q Θpz q
Hθ pz q
1 . KD N COpz qF pz q
1
(5.39)
Assuming a standard second order loop filter, given by: F pz q A0
A1
then Hθ pz q simplifies to: Hθ pz q
z2
pKD A0Ts
zTL z1
(5.40)
pz 1q2 KD A1 Ts2 2qz KD A0 Ts
1
,
(5.41)
which has poles, z0 and z1 , given by:
z0
1 2 A0 KD Ts A1 KD Ts2 2
z1
a
b
KD Ts
A20 KD
A1 pKD Ts p2A0
A1 Ts q 4q
1 2 A0 KD Ts A1 KD Ts2 2 a
b
KD Ts
A20 KD
A1 pKD Ts p2A0
A1 Ts q 4q .
(5.42)
To examine the transient response, the following excitation (or set-point) carrier phase signal is applied to the PLL: θSP rms θStep u pmTs q
1 ωStep pmTs q2 u pmTs q , 2
(5.43)
where θStep is the magnitude of the phase step and ωStep is the magnitude of the frequency step. The response of the PLL to this excitation can be readily found via the inverse z-transform: δθ rms Z 1 tΘSP pz q Hθ pz qu .
(5.44)
Performing a partial fraction expansion prior to inverting the z-transform of (5.44), the ideal PLL response can, readily, be shown to be:
δθ rms
z1m ppz1 1qθStep
ωStep q z0m ppz0 1qθStep z1 z0
ωStep q
.
(5.45)
To compare the PLL across a large range of SNRc conditions, reaching as low 144
Section 5.3: Closed Loop Performance: Linear Operation
TL (s) Disc. KD tA0, A1u z0 z1 β η ζ Bθ (Hz)
High 0.001 Atan 1.0 {16.35,65.10} 0.9912 0.9927 8.06 0.0 1.0 10.0
Low 0.001 Atan 0.4 {16.35,65.10} 0.996-0.00398i 0.996+0.00398i 4.01 -0.996i 0.71 5.22
Comp. 0.001 Atan 0.4 {41.46,165.14} 0.9913 0.9927 8.06 0.0 1.0 10.0
Table 5.1: PLL Design Parameters for Transient Response Experiment
as 0 dB, a relatively narrow PLL filter was chosen. The exact details of the PLL configuration are presented in Table 5.1. The exact magnitude of the transient was chosen, arbitrarily, to be θStep
0.4 rad and ωStep 1.2π rad/s.
Computed
using (5.45), the ideal (noise-free and high-SNRc ) PLL response to this excitation is plotted in Figure 5.11 (c) and labelled ‘High’, along with a simulated response Figure 5.11 (a), also labelled ‘High’. The PLL exhibits a smooth, critically damped response which settles to within 5% of its peak value within 0.5 s. As the purpose of this experiment is to illustrate the impact of KD on the transient response of the PLL, this simulation was repeated under ‘Low’-SNRc conditions. The particular case of SNRc = 0 dB was chosen as it corresponds to KD
0.4
for the arctangent discriminator (see Figure 5.5). An example of the PLL response in this case is shown in Figure 5.11 (a), and labelled ‘Low’. It is evident, apart from the increased noise, that the response of the PLL has become slower and more oscillatory. Inserting the value KD
0.4 into (5.42), the poles of the low-SNRc
system can be evaluated and, subsequently, the decay rate parameter, β and the damping parameter, η, can be deduced. These are presented in Table 5.1. Indeed, β has reduced from a high-SNRc design value of 8.06 to an effective value of 4.01 and η has changed from the critically damped value 0.0 to an underdamped(2) value of
0.996j.
Using KD
0.4 in (5.45), the mean value of this response can be
calculated, and is presented in Figure 5.11 (c) and labelled ‘Low’. The statistics of the low-SNRc response cannot be inferred from an individual trial and, so, this transient was simulated a total of 500 times. The response of the PLL was recorded, the average response was calculated and is presented in Figure 5.11 (b) and labelled ‘Low’. It can be seen that the mean low-SNRc response is, indeed, well approximated by (5.45) using the appropriate value for KD . The design of PLL loop filters is often a delicate balance between a sufficiently fast loop to cope with satellite-to-user dynamics, and a sufficiently slow loop to resist (2)
As discussed in Chapter 4, the z-domain damping coefficient, η, can be related to the traditional damping factor, ζ, used in continuous update loops, via ζ 2 1{p1 η 2 q [104]. In this example, η 0 corresponds to ζ 1.0 and η 0.996j corresponds to ζ 0.71.
145
Chapter 5: The Phase Lock Loop I
δθ (rad)
0.5 0 −0.5 High −1 0
0.5
1 Time (s)
Low
Comp.
1.5
2
(a) One instance of the PLL response.
δθ (rad)
0
−0.2
−0.4 0
High 0.5
1 Time (s)
Low
Comp.
1.5
2
(b) The mean response calculated over 500 trials.
δθ (rad)
0
−0.2
−0.4 0
High 0.5
1 Time (s)
1.5
Low 2
(c) The theoretical response. Figure 5.11: The response of the PLL to phase and frequency steps of -0.4 rad and -1.2 π rad/s, respectively.
146
Section 5.3: Closed Loop Performance: Linear Operation (white) thermal noise induced tracking error. It is crucial, therefore, that a designer have control over the exact placement of the loop poles. To achieve this, the effect of KD must be countered. Examining (5.41), it is clear that the denominator can be rendered independent of KD by scaling the gains, A0 and A1 , by a factor 1{KD . In this manner, the noise-free transient response of the PLL is restored to its high-SNRc response. One common parameter used to quantify the closed loop performance of a tracking loop is the effective rectangular bandwidth of the closed loop transfer function. Denoted here by Bθ , it is defined (similar to that of Bω in Chapter 4), by:
Bθ
1 2πTL
»π
π
|Hθ pejω q|2 dω.
(5.46)
Generally, |Hθ pejω q|2 will be low-pass, with a relatively smooth pass-band. Bθ ,
therefore, is indicative of the speed at which the PLL will settle. In the case of the second order PLL, using the loop filter defined by (5.40) and applying the Cauchy residue method [81, 104], the integral (5.46) admits a closed form solution: 2A1
Bθ
2A K
D A4 K T0 p2A 0 D s 0
A1 KD Ts A1 Ts q
.
(5.47)
Using this expression, Bθ can be calculated for both the high-SNRc case and the low-SNRc cases to be 10.26 Hz and 5.28 Hz, respectively. It is clear that the value of Bθ is significantly reduced by the reduction in KD , an observation which agrees with Figure 5.11 (b) in the sense that reduced Bθ results in an increased mean time to settle. It is noteworthy in (5.47) that the gains A0 and A1 and the gain KD always appear as products (apart from the first term in the numerator, where KD has cancelled). Compensating for the SNRc -induced reduction of KD , by scaling the
filter gains, A0 and A1 , by a factor 1{KD , will therefore restore Bθ to its high-SNRc value. This is illustrated in the final row of Table 5.1 and in Figure 5.11 (b).
It is convenient, here, to denote the high-SNRc value (or the design value) of Bθ by BθDesign and to define it as: BθDesign
Bθ K
D
1 ,
(5.48)
for use in later sections. The parameters of the compensated loop are presented in Table 5.1, in the column labelled ‘Comp.’. An example of the KD compensated loop response is shown in Figure 5.11 (a) and also labelled ‘Comp.’. Again, the mean value of this response is estimated over 500 trials and is plotted in Figure 5.11 (b), once again labelled ‘Comp.’. It is clear that the mean response is restored to that of the highSNRc case. Restoration of the PLL transient performance does, unfortunately, come at a price. It can be seen in Figure 5.11 (a) that the KD -compensated loop response 147
Chapter 5: The Phase Lock Loop I exhibits significantly more thermal noise induced tracking error. This effect, and its implications for PLL design, are considered next.
5.3.2
Thermal Noise Induced Tracking Error and the GNR
Following the transient response of the PLL, once the signal parameters (phase, Doppler and higher order effects) have been estimated, the PLL settles and tracks the carrier phase. This so-called steady-state performance is, typically, dominated by thermal noise. The performance of the PLL in the presence of thermal noise can be measured in terms of the steady-state tracking error variance (often termed 2 . In the case of the PLL, the noise which the tracking jitter), denoted here by σδθ corrupts the estimate θˆ of the carrier phase θ, is nθ , and has propagated through
the discriminator. Similar to the thermal noise floor, N0 , it is convenient to consider an equivalent noise floor for the tracking error estimate e. Denoted here by Nθ , the noise floor of the phase error estimate, in rad2 {Hz, is defined as: Nθ
TL Var
nθ .
(5.49)
Note that, unlike N0 , Nθ is defined as a two-sided PSD. Given the transfer function Hn pz q and (5.49), the tracking error variance can be estimated as:
2 σδθ
Nθ 2πTL
»π
π
|Hn pejω q|2 dω..
(5.50)
Using a similar approach to that of (5.47) and noting the similarity of Hn pz q and Hθ pz q, (5.46) admits the closed from solution for the second order (P
2 σδθ
Nθ 2 TL KD
VarK 2
»π
π
nθ
1) loop:
|Hθ pejω q|2 dω
Bθ
(5.51)
D
From (5.51), the impact of KD on the noise performance of the PLL can be exam2 term in the denominator suggests that σ 2 increases ined. From inspection, the KD δθ
with reduced KD . A more rigorous illustration of this is given in Appendix C. As discussed in Section 5.3.1, it is necessary to compensate for the SNRc -induced
reduction in KD by increasing the filter gains, Ap , by a factor 1{KD . Using such 2 is given by: compensation, σδθ
148
Section 5.3: Closed Loop Performance: Linear Operation
2 σδθ
Var nθ Bθ 2 A A KD A0 Ñ 0 , A 1 Ñ 1
VarK 2
nθ
KD
KD
BθDesign .
(5.52)
D
2 is equal to a conThis result implies that, given perfect KD compensation, σδθ
stant term, BθDesign , divided(3) by the ratio
2 KD Var nθ
r s . As the name suggests, the
constant term, BθDesign , is chosen by the designer. The ratio,
2 KD Var nθ
r s , is related to
SNRc via a function which is particular to each discriminator. It is useful, therefore, to consider this ratio as a metric by which the tracking capability of each discriminator can be compared. This metric, termed the gain-to-noise ratio and denoted by GNR, is defined as: GNR
2 KD Var rnθ s
(5.53)
and expresses the ratio of the square of the discriminator gain to the variance of the discriminator estimate. A plot of GNR for the four-quadrant arctangent discriminator, the arctangent discriminator, decision-directed discriminator and the quadrature discriminator is shown in Figure 5.12. As can be expected from the analysis presented in Section 5.2, under high SNRc conditions (SNRc
¡ 12 dB, for
example), the GNR for each discriminator is quite similar. The reason for this is that KD
1 and Var
nθ
1{ SNRc for each discriminator for high SNRc values. For
reduced SNRc conditions, however, the unique relationship between KD , Var nθ
and SNRc for each discriminator becomes manifest. Because the tracking capability of the PLL can be directly related to the GNR, it provides insight into the relative tracking performance of each discriminator. To investigate the usefulness of the GNR in predicting the relative closed loop performance of various discriminators in the presence of thermal noise, the tracking error variance of a simulated PLL was measured for each of the four discriminators, across a range of SNRc conditions. The loop filter configuration of Table 5.1 was used and perfect KD compensation was applied to the loop filter gains for each case. A total of 29 SNRc conditions were simulated, ranging from -5 dB to 23 dB, for each of the four discriminators. The results of the Monte-Carlo simulations are presented in Figure 5.13. Using (5.52), the theoretically predicted variance was calculated and is also plotted in Figure 5.13, exhibiting good agreement with the simulation results. For SNRc values below approximately 0 dB, the simulation results for the arctangent and decision-directed discriminator have been omitted. In these cases the PLL has (3)
Although the ratio
r s
Var nθ 2 KD
appears in (5.52), its reciprocal is chosen as the definition of the
GNR. This is done so that GNR conforms with metrics such as SNRc , SNRnc and C {N0 , where the numerator pertains to the signal and the denominator pertains to the noise.
149
Chapter 5: The Phase Lock Loop I
Atan2 Atan Sgn(I).Q Q
2
10
GNR
0
10
−2
10
−4
10
−15
−10
−5
0
5 10 SNRc (dB)
15
20
25
30
Figure 5.12: GNR versus SNRc for the four-quadrant arctangent discriminator, the arctangent discriminator, decision-directed discriminator and the quardature discriminator.
lost lock and the resulting measurements of tracking error variance are meaningless. Examining the relative performance of the four discriminators in Figure 5.13, we see that for high SNRc values, all four discriminators perform equally well. For SNRc
8 dB, both of the non-coherent discriminators perform more poorly than
the coherent discriminators. This is to be expected and is the unavoidable cost of achieving insensitivity to data modulation. For the coherent discriminators, the quadrature discriminator slightly outperforms the four-quadrant arctangent discriminator for SNRc
15 dB. When SNRc
10 dB, the difference in performance has increased to 15% and grows rapidly with reducing SNRc from this point. Similarly, for the non-coherent discriminators, the decision-directed discriminator begins to outperform the arctangent discriminator for SNRc
15 dB, the difference becoming significant for SNRc
very low SNRc values, SNRc
10 dB. For
5 dB, the divergence in performance between the
quadrature discriminator and the four-quadrant arctangent discriminator and be-
tween the decision-directed discriminator and the arctangent discriminator begins to reduce. The variance curves tend toward parallel lines (albeit on a logaritmic plot) and settle to a difference of 3.3 dB in both cases. These trends compare well with what is observed in Figure 5.12. In fact, from (5.52), the relative relationship is identical as the curves of Figure 5.13 are simply the reciprocal of the curves of Figure 5.12 multiplied by the constant BθDesign . It is reasonable, therefore, to conclude that a comparison of the relative closed loop tracking performance of PLLs which employ the same loop filter, but different discriminators, can be inferred directly by simply examining the relative GNR of the 150
Section 5.3: Closed Loop Performance: Linear Operation
Atan2 Atan SgnIQ Q
−1
10
σ2δθ (rad2)
−2
10
−3
10
−4
10
−5
0
5
10 SNRc (dB)
15
20
2 versus SNRc for BθDesign 10 Hz and each of four discriminators for both simuFigure 5.13: σδθ lated (markers) and theoretical (solid) results.
discriminators. That is, the relative linear closed loop performance of two PLLs, for any loop configuration, can be inferred by simply examining the open loop behaviour of their respective discriminators.
5.3.3
The Optimum Discriminator for Linear Operation
The ratio of the GNR of the four-quadrant arctangent discriminator to the quadrature discriminator and the ratio of the GNR of the arctangent discriminator to the decision-directed discriminator for a range of SNRc conditions is shown in Figure 5.14. This plot, in conjunction with the linear region plot of Figure 5.10, can 2 for any given loop filter be used to choose a discriminator which will minimize σδθ
choice. The choice of discriminator can be considered for two different discriminator classes, namely coherent and non-coherent.
For the coherent discriminators, under high SNRc conditions (¡ 11 dB) the four-
quadrant arctangent discriminator incurs less than a 10% performance degradation,
when compared with the quadrature discriminator, yet it exhibits a significantly larger linear region. The four-quadrant arctangent discriminator should, therefore, be used in this region as it provides more robustness than the the quadrature discriminator, being capable of absorbing larger phase transients while maintaining linear operation.
In the region 3 SNRc
11 the optimum choice of discriminator is dependent 151
Chapter 5: The Phase Lock Loop I
1
GNR Ratio
0.9
GNRAtan2/GNRQ GNRAtan/GNRSgn(I).Q
0.8
0.7
0.6
0.5 −10
0
10 SNRc (dB)
20
30
Figure 5.14: The ratio of the GNR of the four-quadrant arctangent discriminator to that of the coherent discriminator and that of the arctangent discriminator to that of the decision directed discriminator for a range of SNRc values.
on the application, the quadrature discriminator significantly outperforms the fourquadrant arctangent discriminator but has a notably narrower linear region. For application where low tracking error is the main priority, the quadrature discriminator should be used whereas, if resilience to signal dynamics is desired, a designer may wish to avail of the larger linear region of the four-quadrant arctangent discriminator.
For very low SNRc values ( 3 dB) the linear regions of both discriminators are
similar, yet the quadrature discriminator provides approximately 3 dB less tracking error variance and should, therefore, be used. Unlike the coherent discriminators, the choice of discriminator is simpler for the non-coherent discriminators. At SNRc
9 dB the linear regions of the arctan-
gent discriminator and the decision-directed discriminator begin to converge. Also for reducing SNRc values around this point, the GNR of the decision-directed discriminator begins to significantly outperform the arctangent discriminator. Thus, for SNRc values above approximately 9 dB, the arctangent discriminator should be used while, for SNRc values below this point, the decision-directed discriminator should be employed. These conclusions are presented graphically in Figure 5.15. It is worth commenting on one (perhaps obvious) conclusion: as the performance of the coherent discriminators, both in terms of linear region and GNR, is superior to that of the non-coherent discriminators, a coherent discriminator should always be used for steady-state, linear operation, where possible. That is, when the re152
Section 5.4: Closed Loop Performance: Non-Linear Operation
Coherent
Q
Non−Coherent
Atan2: Dynamics Q: Accuracy
Sgn(I).Q
−10
0
Atan2
Atan
10 SNRc (dB)
20
30
Figure 5.15: Preferred choice of PLL discriminator for the coherent and non-coherent cases.
ceived signal is not modulated by a data stream (for example, the GIOVE-A B/C and GIOVE-B B/C pilot channels) or in cases where some data wipe-off scheme is available, (through, for example, some A-GPS system), then a coherent discriminator will provide superior performance when the PLL is operating in its linear region. This conclusion, however, does not extend to the non-linear operation of the PLL, as will be discussed next. It is noteworthy that whilst the analysis presented here considered only four discriminators, the metrics and the theoretical model employed (KD , GNR and LR), can be extended to consider, and provide a comparative analysis of, a host of carrier phase estimators. Given an expression for these three metrics as a function of SNRc , this analysis could be extended to consider any new memory-less discriminators. Finally, as has been shown in Section 5.2, the expressions for the gain and variance of the arctangent based discriminators, to the best of the author’s knowledge, do not yield simple closed form solutions. Calculation of the various discriminator parameters would, therefore, require the use of numerical integration techniques. As an alternative to numerical integration, a set of simple approximate functions for the gain and variance of both the four-quadrant arctangent and the arctangent discriminators are presented in Appendix B.
5.4
Closed Loop Performance: Non-Linear Operation
The previous sections have considered the linear operation of the PLL. However, thus far, the question of how the PLL settles to this linear condition, has been 153
Chapter 5: The Phase Lock Loop I neglected. Generally, prior to this use of a PLL, the receiver will refine the coarse carrier frequency estimate, provided by the signal acquisition algorithms, using an FLL(4) . Upon initialisation of the PLL, the refined carrier frequency estimate is applied to the PLL NCO. As the FLL is a phase-incoherent device, the initial carrier phase error is in the range
rπ, πs.
The PLL begins to estimate the carrier phase
error and pursues the carrier phase along a trajectory corresponding to the initial carrier frequency and drift errors. This process is known as the PLL acquisition stage [79] and, if successful, will result in the PLL converging to zero mean carrier phase and carrier frequency errors. Typically, a loop filter suited to the task of acquisition will be used for this stage and, once the PLL has settled, a different loop filter, more suited to steady-state tracking, is employed (the latter will be discussed further in Chapter 6). This section examines this acquisition stage and seeks to discover the influence of the carrier phase discriminator on the acquisition performance. The performance is assessed on the basis of the time taken for the PLL to settle, denoted TTS (time to settle). The exact details of this metric will be presented in Section 5.4.2. As the initial carrier phase error may not, necessarily, lie within the linear region of the carrier phase discriminator, it is unlikely that the PLL will reside in its linear region of operation for the duration of this process. Furthermore, the PLL is subject to both deterministic step frequency and phase errors and the stochastic influence of thermal noise. The linear theory presented in the previous section cannot, therefore, be applied. Rather than resort to a potentially unwieldy non-linear transient analysis of the acquisition stage, (the aim, after all, is simply to perform a comparative analysis of the influence of the discriminator function on the settling time), insight is sought through the use of Monte-Carlo simulation. By examining a selection of representative examples, some conclusions are drawn, regarding the preferred choice of carrier phase discriminator(5) .
5.4.1
The PLL Acquisition Stage
At the instant that the PLL is closed, there normally exists residual carrier phase, carrier frequency and drift errors. By recursively estimating the phase error, the PLL generates an estimate of the signal phase, frequency and, depending on its order, the frequency drift. As the carrier frequency is not estimated directly, the PLL may experience a sustained frequency error for a substantial duration. This frequency (4)
Of course, the coarse code phase estimate must also be refined, but this problem is not considered here. It is assumed that the DLL is ideal, and operating with zero tracking error. In reality, when the DLL is non-ideal, the corresponding synchronisation errors manifest themselves as a degradation of SNRc . (5) An interesting mathematical analysis of a simplified model of the PLL, assuming a continuous update loop and a quadrature discriminator, is conducted in [79]. Utilising the Fokker-Planck technique, analytical results of the acquisition probabilities of this loop model are presented. Owing to the nonlinear nature of the discriminator functions described here, in Section 5.2, such an analysis is not attempted in this study.
154
Section 5.4: Closed Loop Performance: Non-Linear Operation error will cause the phase error to increase (or decrease) and may result in a phase error which is outside the linear region of the carrier phase discriminator. In fact, this error may exceed some full carrier phase periods. Being periodic in carrier phase, the discriminator will produce error estimates which drive the PLL to the nearest stable lock point. These lock points are the points where the mean discriminator response curves intersect the x-axis with positive slope. For the coherent discriminators,
n2π (see, for example, Figure 5.4), and for the non-coherent discriminators they are δθ nπ, n P N (Figures 5.7 and 5.8). Because of their
these are given by δθ
ability to provide synchronisation both in phase and in anti-phase with the carrier,
the non-coherent discriminators have twice as many stable lock points as the coherent discriminators. The net effect of this periodicity, in terms of PLL acquisition, is that the PLL may not synchronise with the carrier at the nearest lock point but, due to the carrier frequency and drift errors, may traverse a number of half or full carrier periods before locking. The initial residual frequency error, denoted here by δω0 , plays a significant role in the acquisition process. As the FLL is subject to AWGN, it is not surprising that the frequency error is normally distributed. As shown in Chapter 4, the error has zero mean and its variance is proportional to the FLL bandwidth, Bω , and inversely proportional to the prevailing SNRc . The exact variance can be calculated using (4.13), examples of which are presented in Figure 4.14. The initial drift rate, denoted δ ω9 0 , discussed in Chapter 2, is slowly varying and takes on a value in the range 0.9
to 0 Hz/s. For the purposes of PLL acquisition, it can be adequately modelled as a uniform random variable which retains a constant value for the duration of the acquisition. The initial carrier phase, denoted δθ0 , is not estimated by the FLL and so is modelled as a random variable, uniformly distributed between
π and π.
The PLL can reach its settled state through a number of distinct processes, some of which are depicted in Figure 5.16. The most simple is that of direct acquisition, depicted in Figure 5.16 piq and piiq. In this process, the PLL behaves in an approx-
imately linear fashion and draws the local phase directly toward the nearest stable lock point. The second case is depicted in Figure 5.16 piiiq and represents the case
where, owing to the initial carrier phase and frequency errors, the discriminator drives the phase toward one stable lock point (in this case 0 rad) while the initial carrier frequency error draws the PLL to another lock point (in this case π rad). An apparent stalling of the PLL is observed which results in a prolonged settling time and is the result of the PLL exercising the non-linearity of the discriminator. A similar effect occurs when the initial carrier phase error assumes a value such that the discriminator is in a low-gain region (for example δθ
2.0 rad in Figure:5.9).
The PLL begins to slowly converge to the nearest lock point (owing to the low gain)
and, as the carrier phase error reduces, the phase error returns to the linear region, and as the gain increases, it begins to converge more rapidly. Both of these non-
linear effects result in increased settling time. Figure 5.16 piv q and pv q represent 155
Chapter 5: The Phase Lock Loop I
8 6
TTS = 0.558 s 4 2
TTS = 0.332 s
δθ (rad)
0
TTS = 0.205 s
−2
(i) (ii) (iii) (iv) (v)
−4 −6
TTS = 0.713 s −8 −10 −12 0
0.2
0.4
0.6
0.8
1
Time (s) Figure 5.16: Examples of the carrier phase error throughout the PLL acquisition process. The solid lines represent the final stable lock points and the broken lines represent the extremes of region, from nπ θT h to nπ θT h .
156
Section 5.4: Closed Loop Performance: Non-Linear Operation the case where the initial frequency error is sufficiently large that the PLL does not settle to the nearest stable lock point, instead it is driven away from this point in the direction of the initial carrier frequency error. The carrier phase error grows and, as the discriminator is periodic, at a certain point the discriminator error will change sign and drive the PLL away from the first stable lock point and towards the next. Depending on the relative magnitude of the initial carrier phase and carrier frequency errors, and the PLL bandwidth, the PLL may settle to this stable lock point or it may overshoot it and be drawn towards the next. If the frequency error is sufficiently large, or the PLL bandwidth is sufficiently low, the PLL may simply fail to lock. The maximum frequency error that the PLL of a given bandwidth can sustain is often referred to as the PLL pull-in range [79].
5.4.2
Time To Settle (TTS)
The duration of time from the instant that the PLL is closed to the instant that the PLL has settled to its steady state value is known as the settling time. There is no absolute measure of the instant that the PLL is settled and the definition of the settled state is quite subjective. Indeed, no PLL is perfect and none will settle to a zero phase error. Here, a threshold, denoted θT h , is chosen and the time to settle defined as the point at which the carrier phase error falls within, and remains within, the region: n2π θT h
δθ n2π
θT h ,
(5.54)
for the coherent discriminator, and the region: nπ θT h
δθ nπ
θT h
(5.55)
for the non-coherent discriminator (again, for n P N). The choice of θT h , therefore,
defines TTS and should be chosen such that TTS accurately reflects the performance of a given PLL design. The value of θT h should be small enough that TTS indicates the instant where the PLL is truly settled, but not so small as to unfairly penalise systems which exhibit a small steady-state error. Also, as the PLL is subject to AWGN, θT h should be large enough that it is not breached by spo-
radic noise-induced phase excursions. That is, a PLL that is adequately settled will exhibit a stochastic tracking error which may, with small but non-zero probability, exceed θT h . A sufficiently large value of θT h will ensure that this probability, and the probability of the resulting corrupted TTS measurement, is negligible. The range of mechanisms which result in a settled PLL state result in a range of corresponding TTS values. Some of these can be illustrated by assuming, for
π{4, and considering the carrier phase transients of Figure 5.16. The response labelled piq begins with n2π θT h δθ0 n2π θT h and with δω
example, θT h
157
Chapter 5: The Phase Lock Loop I sufficiently small such that θT h does not breach the threshold for the duration of the settling process. The TTS value corresponding to all such settling mechanisms is 0 s.
The transient labelled piiq also begins with n2π θT h
δθ0 n2π
θT h but
does breach the threshold and does not return until a time of approximately 0.205 s. The transient labelled piiiq begins below the settled region and converges toward
its final stable lock point, overshoots and breaches the settled region on the upper side before finally settling with a TTS of approximately 0.332 s. Transients which settle with this mechanism tend to have relatively low settling times. Empirically, it has been found that there is a high probability that such mechanisms will result in a low settling time, as illustrated in Figure 5.17.
The settling mechanisms of the transients labelled piv q and pv q traverse at least
one settling region before converging to their final stable lock points. It has been found that this mechanism results in settling times with a Gaussian-shaped distribution around a non-zero mean. As the initial conditions of the PLL (δθ0 , δω0 and δ ω9 0 ) are random variables and the PLL is subject to AWGN, the resulting TTS is also a random variable. To gain insight into the TTS and the influence of the discriminator it, the distribution of TTS over a large number of Monte-Carlo trials is examined. An example of the
distribution of the settling time (denoted f pTTSq) is shown in Figure 5.17. The distribution was estimated by examining 10, 000 Monte-Carlo simulations using a
10 Hz and a four-quadrant arctangent discriminator, assuming a prevailing carrier-to-noise ratio of C {N0 48 dB Hz. The initial PLL with a bandwidth of Bθ
carrier frequency error was assumed to be a Gaussian random variable with zero
mean and variance of 440 rad2 s2 , both the initial drift rate and carrier phase were assumed to be uniformly distributed, as discussed earlier. It can be seen that the distribution exhibits an impulse at TTS
0 s, resulting from settling mechanisms
similar to that of Figure 5.16 piq. Adjacent to this impulse, the distribution decays
exponentially. It is proposed that this feature is a result of such settling mechanisms as that of Figure 5.16 piiq and piiiq. Finally, centred around TTS 0.3 s, the den-
sity is approximately Gaussian, a feature which, it is proposed, results from settling mechanisms similar to those of Figure 5.16 piv q and pv q.
While the distribution of TTS is, perhaps, useful in discerning the relationship between settling mechanism and settling time, it offers too much raw information to be useful in a comparative analysis of the influence of the carrier discriminator. Reducing the distribution to a single metric facilitates a direct comparison of different discriminators. The question of what single metric to use is, of course, arbitrary, but may significantly influence the comparison; it must, therefore, be chosen carefully. Recall that the purpose of this study is to ascertain the influence of the carrier discriminator on the settling time. In particular, the aim is to minimise the settling time. As the process is stochastic, it can never be guaranteed that a particular PLL 158
Section 5.4: Closed Loop Performance: Non-Linear Operation
8 Type (i)
TTS90% = 0.343 s
7 6
f(TTs)
5 4
Type (ii) and (iii)
Type (iv) and (v)
3 2 1 0 0
0.1
0.2
0.3
0.4 0.5 TTS (s)
0.6
0.7
0.8
Figure 5.17: Example pdf of TTS for the four-quadrant arctangent discriminator for a C {N0 48 dB Hz and Bφ 10 Hz. The influence of each type of settling mechanism, as depicted in Figure 5.16, on the distribution of settling times is annotated, as is the value of T T S90% .
159
Chapter 5: The Phase Lock Loop I acquisition has completely settled within an particular period of time. Rather, the probability that the PLL has settled within a certain period of time can be defined. Moreover, a time can be defined, denoted here by TTSx% , within which there is an x% probability that the PLL has settled. Similar to the cumulative density function, TTSx% is formally defined as:
TTSx%
$ &
%T P R :
»T
f ptq dt
, .
x , 100 -
(5.56)
0
where f ptq denotes the probability density function of the settling time, as depicted
in Figure 5.17. Interestingly, if a high probability is chosen, for example 90% or 95%, this metric can be employed to schedule the transition to a PLL tuned for steady-state performance. Another reason for choosing TTSx% as a performance metric is that it can be
readily calculated without full knowledge(6) of f ptq. Form a practical standpoint, it
is not feasible to execute a simulation indefinitely, until such time as the PLL settles and a value of TTS can be recorded. Indeed, in some instances, the PLL may not settle at all(7) . To calculate TTSx% using (5.56), knowledge of f ptq for very large values of t is not required. For example, calculation of TTS90% in Figure 5.17, only
requires knowledge of f ptq in the range 0 t 0.4 s. If a reasonable upper limit on
the simulation duration is chosen, for example 15 s, then, for most reasonable values
of TTSx% , the required region of f ptq can be estimated. Note, however, higher values
of x, require more knowledge of f ptq and, thus, require longer simulation durations.
5.4.3
The Experiment
To compare the performance of each of the four carrier phase discriminators discussed in Section 5.2, values of TTSx% were calculated for a selection of representative PLL designs under a range of operating conditions. A simulation configuration consisted of a particular PLL design and a certain operating condition. Each simulation configuration was applied to 10, 000 Monte-Carlo trials. The settling time for
each trial was recorded and the probability density function, f pTTSq, was estimated
from these 10, 000 settling times. Finally, TTSx% was calculated using (5.56) and
this estimate of f pTTSq. The range of simulation configurations consisted of all possible combinations of: (6)
This is in stark contrast to metrics such as the mean or variance, or other moments which require full knowledge of the corresponding distribution. (7) For the particular simulation trials conducted in this study, it was found that the vast majority of trials settled within 15 seconds. Occasions where the PLL did not settle within this period were very infrequent. The lowest probability of settling within 15 seconds observed was for the decisiondirected discriminator (Sign(I).Q) using a 10 Hz loop bandwidth and a carrier-to-noise-floor ratio of 33 dB Hz. In this case, fifty of the 10, 000 trials did not settle. Also, for the same configuration, using the arctangent discriminator, thirty of the 10, 000 trials did not settle.
160
Section 5.4: Closed Loop Performance: Non-Linear Operation Fs ωRF ωIF h0 h1 h2 PRN
5.489 1575.42 2π 1.266 2π 3.9 1022 2.4 1021 2.4 1022 GPS C/A #1
(M Hz) (M rad/s) (M rad/s)
Table 5.2: Receiver configuration for TTS simulations.
each of the four carrier phase discriminators: the four-quadrant arctangent,
the arctangent, the decision-directed quadrature and the quadrature discriminators. each of eight loop filters having critical damping (η
0) and a two-sided
equivalent rectangular bandwidth (Bθ ) of 10, 12, 15, 20, 25, 30, 35 and 40 Hz, respectively. each of six carrier-to-noise-floor ratios (C {N0 ) of 48, 45, 42, 39, 36 and 33 dB Hz,
respectively..
The initial conditions, δθ0 , δω0 , δ ω9 0 were randomly generated according to:
U pπ, πq δω0 N p0, 440q δ ω9 0 U p0.2, 0.8q . δθ0
(5.57) (5.58) (5.59)
The remaining PLL configurations are detailed in Table 5.2 and the following assumptions were also made: no multi-path propagation effects were present, the user was stationary, all signal dynamics were induced by satellite motion
and the local oscillator, the code tracking loop provided perfect code wipe-off, neither RF interference nor co-channel interference were present in the received
signal. Finally, it was assumed that the receiver could perfectly estimate the prevailing SNRc such that the discriminator normalisation, described in Section 5.3.1, could be applied to the discriminators. This is deemed a reasonable assumption as the receiver would, by necessity, have tracked the signal with the FLL for sufficient time to refine the carrier frequency estimate. This duration is, generally, sufficient to also produce reasonably accurate carrier power and noise-floor estimates. 161
Chapter 5: The Phase Lock Loop I A total of 192 simulation configurations were assessed, with each configuration applied to 10, 000, fifteen second Monte-Carlo simulations(8) . Each of the 10, 000 simulations were configured with a randomly generated set of initial signal conditions, δθ0 , δω0 and δ ω9 0 . For each simulation, the value of TTS was recorded, assuming a threshold of θT h
5.4.4
π4 .
The Results
The results of the simulations were processed to estimate each of the 192 TTS distributions, corresponding to each of the combinations of C {N0 and Bω pairs and
each of the four discriminators. The settling probability was chosen to be 90%, a compromise between two conflicting requirements. 90% is deemed sufficiently high that it is representative of the majority of the settling mechanisms. For example,
0.18 s and would effectively exclude the settling mechanisms of Figure 5.16 , piv q and pv q. Conversely, 90% is sufficiently low that it is insensitive to uncertainty of f ptq for examining Figure 5.17, a 60% probability would result in TTS60%
high t values. Recall that the maximum simulation duration has been limited to
15 s and the number of trials per distribution limited to 10, 000. As the distribution tapers for high t, less simulation points are used to estimate the higher portion of
the distribution and, thus, it is subject to a higher esimation error. Moreover, f ptq is
absolutely unknown for t ¡ 15 s. The calculation of TTS90% , utilizes only the lower portion of f ptq and, therefore, achieves higher accuracy than, for example, TTS99% .
Figure 5.18 presents sixteen of the 192 estimated TTS distributions, representing
each of the four discriminators for both the highest and lowest C {N0 conditions (48
and 33 dB Hz) and the highest and lowest loop bandwidth (40 and 10 Hz). Also
shown are the TTS90% estimates for each of the distributions. Rather than examine each of the 192 results, the data can be reduced somewhat by plotting TTS90%
against Bθ for a selection of C {N0 values. This is illustrated in Figure 5.19 for three
of the six C {N0 values: 48, 45 and 33 dB Hz.
5.4.5
Preferred Discriminator for the Acquisition Stage
Caveat lector : Before drawing any conclusions on the influence of various design parameters on TTS90% , it is important to recall that this has been a numerical study. While the simulation configurations considered here are representative of a range of typical receiver operating conditions, observations and conclusions based on the outcomes of these simulations cannot be universally applied. Nonetheless, this study aims to identify the features of the receiver design which most heavily influence the settling time, with a view to improving subsequent heuristic design procedures. (8)
In total, 28, 800, 000 seconds (or 333 days 8 hours) of simulation time was executed. The simulations were executed on a 10 node cluster, each node consisting of eight 2.5GHz cores, with 8 GB of RAM [25].
162
Section 5.4: Closed Loop Performance: Non-Linear Operation
Atan2
Atan
Sgn(I).Q
Q
f(TTs)
400 300 200 100 0 0
0.01
0.02
(a) C {N0
0.03 TTS (s)
0.04
0.05
48 dB Hz, Bθ 40 Hz
f(TTs)
200 150 100 50 0 0
0.01
0.02
(b) C {N0
0.03 0.04 TTS (s)
0.05
0.06
33 dB Hz, Bθ 40 Hz
f(TTs)
10
5
0 0
0.2
0.4
0.6 0.8 TTS (s)
(c) C {N0
1
1.2
48 dB Hz, Bθ 10 Hz
8 f(TTs)
6 4 2 0 0
0.5
1
(d) C {N0
1.5 TTS (s)
2
2.5
33 dB Hz, Bθ 10 Hz
Figure 5.18: An example of sixteen estimated TTS distributions for each of the four carrier phase discriminators and for each of two loop bandwidths (Bθ 10 and 40 Hz) and each of two carrier-to-noise-floor ratios (C {N0 48 and 33) dB Hz. The vertical broken lines represent the corresponding value of TTS90% .
163
Chapter 5: The Phase Lock Loop I
Atan2
0
Atan
SgniQ
Q
TTS (s)
10
−1
10
−2
10
10
15
20 (a) C {N0
25 Bθ (Hz)
30
35
40
25 Bθ (Hz)
30
35
40
25 Bθ (Hz)
30
35
40
48 dB Hz
0
TTS (s)
10
−1
10
−2
10
10
15
20 (b) C {N0
42 dB Hz
0
TTS (s)
10
−1
10
−2
10
10
15
20 (c) C {N0
33 dB Hz
Figure 5.19: Monte-Carlo simulation based estimates of TTS90% versus Bθ for each of the four carrier discriminators for three carrier-to-noise ratios.
164
Section 5.4: Closed Loop Performance: Non-Linear Operation Moreover, rigorous justification through simulation of certain propositons may serve to bolster a design procedure, which would otherwise be based entirely on intuition. In particular, conclusions drawn from this study are limited to PLLs with (twosided) bandwidths in the range 10 to 40 Hz, to carrier-to-noise-floor ratios in the range 33 to 48 dB Hz and to initial carrier frequency errors that are normally
distributed with a variance of 440 rad2 s2 . Extrapolations of conclusions drawn from this finite study must be made with care as they may result in a sub-optimal design. Examining the 192 estimated distributions (sixteen of which are presented in
Figure 5.18) it is evident for a given combination of Bθ and C {N0 , that the value
f p0q for the non-coherent discriminators (Atan and Sign(I).Q) is, generally, higher
than for the coherent discriminators (Atan2 and Q). Exceptions to this observation are cases with low Bθ and C {N0 values. As indicated in Figure 5.17, an impulse at T T S
0 s is attributable to the settling mechanism of Figure 5.16 (i), where
the carrier phase error begins within, and remains within, the settled region. It is proposed that the reason for this is simply that the non-coherent discriminators have stable lock points spaced every π rad as opposed to 2π rad for the coherent discriminators. More stable lock points implies a higher likelihood that the initial carrier phase will lie within the settling region. The presence of this impulse at TTS 0 s results in a significant reduction of TTS90% . Another trend that is evident
is that the normally distributed settling mechanisms, labelled as Type piv q and pv q in
Figure 5.17, is less prominent in the non-coherent discriminator distributions than in the coherent discriminator distributions. Again, this results in an overall reduction in TTS90% . Interestingly, the settling time of the coherent discriminators appears to be rel-
atively insensitive to changes in C {N0 (within the range examined here), unlike the
non-coherent discriminators, which show a stark increase in TTS90% for decreasing
C {N0 . This, it transpires, has implications for the preferred choice of discriminator. Examining the relationship between Bθ and TTS for each of the six C {N0 cases
(three of which are depicted in Figure 5.19) it is not immediately evident that
one discriminator should be the preferred choice. It is clear, however, that the quadrature discriminator achieves TTS90% values which are consistently higher than the best performing discriminator, across all tested configurations. Similarly, the decision-directed quadrature discriminator (Sign(I).Q), never achieves the lowest TTS90% value for any given configuration and, for low Bθ and C {N0 , it performs
considerably worse than the best discriminator. It is suggested, therefore, that neither the quadrature nor the decision-directed quadrature discriminators are ever the preferred discriminator. Either the arctangent or the four-quadrant arctangent discriminator will outperform these two discriminators for any of the Bθ and C {N0 configurations considered here.
As shown in Section 5.3.3, the choice of discriminator which minimises tracking 165
Chapter 5: The Phase Lock Loop I error variance for linear PLL operation, is dependent on the presence or absence of data modulation. In the absence of data modulation, the discriminator choice which minimises tracking error variance is either the quadrature discriminator or the four-quadrant arctangent discriminator, depending on the prevailing SNRc . In the presence of data modulation, neither of these coherent discriminators can be used and tracking performance must be sacrificed to achieve insensitivity to data modulation and, again depending upon SNRc , either the decision-directed quadrature discriminator or the arctangent discriminator must be used. Similarly, for the PLL acquisition stage, the restrictions of insensitivity to data modulation restrict the choice of discriminator. Considering, first, PLL design in the presence of data modulation: a choice between the two non-coherent discriminators is simple, the arctangent always outperforms the decision-directed discriminator and, thus, should always be employed. For data-less signals, however, the choice is more involved. Upon inspection of the arctangent, or the four-quadrant arctangent discriminator, TTS90% curves it is clear that the arctangent discriminator achieves the lowest TTS90% values for high Bθ and the four-quadrant discriminator achieves TTS90% values for low Bθ . Surprisingly, even for data-less signals, the choice is between one coherent discriminator and one non-coherent discriminator. The Bθ point at which the two TTS90% curves intersect defines the threshold at which the preferred discriminator changes from being the four-quadrant arctangent discriminator to the arctangent discriminator. From Figure 5.19 it is clear that this Bθ threshold is not
fixed, in particular, it exhibits a significant variation with changes in C {N0 , changing from approximately Bθ C {N0
12.7 Hz at C {N0 48 dB Hz to Bθ 25.8 Hz at
33 dB Hz. The full set of six threshold points, corresponding to each of the simulation C {N0 values, is plotted in Figure 5.20, labelled ‘Equal TTS90% ’(9) . The figure can be interpreted as follows: to minimise TTS90% for the PLL acquisition
stage for a data-less signal, the four-quadrant discriminator should be used when the design choice of C {N0 and Bθ , corresponds to a point in the tC {N0 , Bθ u plane
which lies below the curve of Figure 5.20. Otherwise, the quadrant discriminator should be employed. It has been shown here that the preferred choice of discriminator for the PLL acquisition stage may differ from that of the optimum discriminator choice for steady state tracking. Having employed one particular discriminator in the acquisition stage, once the PLL has settled, the discriminator should be changed to the optimum choice for steady state operation, as detailed in Section 5.3.3. This poses no problems for the non-coherent PLL design, as the preferred discriminator for the acquisition stage is non-coherent and bears the same stable tracking points as the optimum non-coherent discriminator for the steady state tracking stage (δθ
nπ).
(9) Note that the exact values of the thresholds were found by linear interpolation of the data points of Figure 5.19
166
Section 5.4: Closed Loop Performance: Non-Linear Operation This is not the case for the data-less signal. When the prevailing C {N0 and cho-
sen Bθ dictate the the four-quadrant discriminator is used, following the settling of the PLL, the discriminator can simply be interchanged with whichever of the coherent discriminators is optimal, as both exhibit the same stable lock points (δθ
n2π).
If the arctangent discriminator has been employed for acquisition, however, the discriminator cannot simply be interchanged with the optimal coherent discriminator, once the PLL has settled. The reason for this is that the arctangent discriminator may have settled in anti-phase with the received signal (at one of the stable lock points
3π, π, π, 3π...). Before transitioning from a non-coherent discriminator to
a coherent discriminator, the sign of the in-phase correlator value, Im , should be
estimated. If it is positive then the local carrier and the received signal are in phase and the PLL can be closed using the coherent discriminator. If it is negative, the local carrier should be incremented (or decremented) by π radians prior to closing the PLL with the coherent discriminator. Of course, a test of the sign of Im is subject to error, as Im is corrupted by AWGN. Calculating the associated probability of incorrectly identifying the sign of Im is tantamount to calculating the BER of a NRZ BRSK modulation scheme. Denoted PErr , this probability is given by [94]: PErr
21 erfc
a
TL C {N0 cos pδθq .
(5.60)
When evaluating (5.60), it is clear that as the PLL has begun to settle, then
δθ
π 4,
and, thus, cos pδθq
¡
0.707. For high C {N0 values, PErr is quite low
(approximately 9.0 109 for C {N0
42 dB Hz), for lower C {N0 values, this
probability becomes considerable (approximately 0.023 for C {N0
33 dB Hz).
It
can be readily reduced, however, by observing Im for more than one integration period, as there is no data modulation present (effectively increasing TL ). This added observation period, perhaps, should be considered in the total measure of TTS90% , as it represents more accurately the point at which the PLL is ready to
commence steady state tracking. For high C {N0 values, this added burden is simply one integration period (1 ms for the GPS L1 C/A signal) and does not alter the previous conclusion on what choice of discriminator should be used. For low C {N0
values, examining Figure 5.19, it can be seen that the added burden of estimating the sign of Im over numerous integration periods may render the use of the arctangent discriminator less desirable than simply using the four-quadrant discriminator. This is particularly likely as the difference in TTS90% for these two discriminators is quite
small for high Bθ values and low C {N0 values, differing by less than 5 ms for Bθ
Hz in Figure 5.19 (c).
¥ 30
If, for example, a probability of error is stipulated, and the number of integration periods chosen accordingly, to satisfy this probability of error via (5.60), and this observation time is added to TTS90% for the arctangent discriminator, then a 167
Chapter 5: The Phase Lock Loop I
40 35
PErr < 1.0×10−12
Equal TTS90% 30
Use Atan
Bθ (Hz)
25 20 15 10
PErr < 1.0×10−6
5 0
Use Atan2 32
34
36
38
40 42 C/N0 (dB)
44
46
48
Figure 5.20: A plot of the intersection points of TTS90% versus Bθ curves for the arctangent and four-quadrant arctangent discriminators for each of the C {N0 conditions and of the modified intersection points when the overhead of bit estimation for both PErr 1.0 106 and PErr 1.0 1012 are considered.
new threshold can be defined. This threshold defines the region for which the arctangent discriminator outperforms the four-quadrant discriminator, accounting for the added overhead of bit estimation. Choosing the maximum probability of error and calculating the required observation period for each case of Figure 5.19, two new thresholds are defined and depicted in Figure 5.20, corresponding to maximum
probabilities of error of 1.0 106 and 1.0 1012 . Again, the optimum discrimi-
nator is simply found by examining the region in which the point tC {N0 , Bθ u lies, as illustrated in Figure 5.20.
5.5
Conclusions
This chapter began with a simple, linear, mathematical model of the performance of the discrete update PLL. To enhance this model, a characterisation of four carrier phase discriminators in terms of mean, gain, variance, linear region, and a novel performance metric, GNR, was conducted. This characterisation identified the issue of SNRc -induced gain degradation prevalent in some of these discriminators. The related performance degradation that is observed was highlited and a simple measure countering this effect was presented. The closed loop noise performance of the PLL, when operating in its linear 168
Section 5.5: Conclusions region, was considered and a comparison of the performance of each of the four discriminators was presented. It was illustrated that the GNR can prove a useful metric which can be used to infer the relative closed loop performance of various discriminators, based on their respective open loop characteristics (namely, KD and
Var nθ ). Utilizing both this metric and the linear region analysis, guidelines for optimal discriminator choice are presented, wherein it is shown that the discriminator choice should be based upon desired application and the prevailing SNRc . A Monte-Carlo based numerical analysis of the closed loop performance of the PLL throughout the PLL acquisition stage was then performed. Specifically, the PLL was assessed in terms of the time taken to settle. It was shown that, to minimise the time taken to settle, the discriminator choice should be based on the prevailing C {N0 and the design PLL bandwidth. Of the four discriminators examined, it was
found that the choice reduces to two discriminators. A simple design routine was presented that can guide a designer to the preferred discriminator choice, given the choice of Bθ and the prevailing C {N0 .
This analysis has considered both the open-loop characteristics of the PLL dis-
criminators, and their respective performance in both the linear region of operation and under nonlinear operating conditions. While the selection of carrier discriminator has been considered, the design of the loop filter, F pz q, has, so far, been
neglected. Moreover, the effects of deterministic carrier phase and frequency variations and of thermal noise have been addressed here, yet the effects of stochastic phase anomalies have been neglected. In particular, local oscillator imperfections can have a considerable influence on PLL performance. The next chapter, therefore, considers the problem of loop fitter selection for the PLL, conditioning the loop design on the influence of thermal noise and of oscillator phase noise.
169
170
Chapter 6
The Phase Lock Loop II: Loop Filter Design Conditioned on Thermal Noise and the Local Oscillator Having considered the performance analysis of the carrier phase discriminator and its influence on both the transient and steady state closed loop performance of the PLL in Chapter 5, the focus is now turned to the loop filter design. In particular, while Chapter 5 has assessed the problem of tracking deterministic phase processes (such as phase and frequency steps and the acquisition process) in the presence of thermal noise, this chapter will study the problem of tracking stochastic phase processes. Utilizing a standard power spectral density model of the receiver’s local oscillator, the oscillator phase process is described as an ensemble of random walks of various orders. This model is employed in an optimisation of the loop filter which strives 2 , of a PLL subject to to minimise the net carrier phase tracking error variance, σδθ
both thermal noise interference and a non-ideal local oscillator. The minimisation of the tracking error (that is, the MMSE) can be expressed as a problem of optimal filtering. Here, owing to the stochastic nature of the error sources, it is expressed as a Wiener filtering problem [22]. The purpose of a Wiener filter is to estimate the signal content of a noisy measurement. It is assumed that both signal and noise are stochastic in nature and, in general, have overlapping spectra. The Wiener filter, in various causal and anti-causal forms, is the MMSE filter, smoother or predictor of a signal in noise. Its application, as a causal filter, to the problem of carrier phase tracking yields simple expressions for stable, optimal (in the MMSE sense) PLL filters. The performance of this filter is examined and the validity of some simplifying assumptions made in its development are assessed through simulation and real171
Chapter 6: The Phase Lock Loop II signal experimentation. Finally, the application of such a filter design to adaptive loop implementation is explored and it is shown that an adaptive PLL based on this Wiener filter has significant advantages over fixed bandwidth PLLs.
6.1
The Local Oscillator
Ideally, a reference oscillator would emit a sinusoidal voltage at a single, fixed frequency. In practice, however, this is never the case and reference oscillators can produce a wide range of waveforms with a time-varying fundamental frequency. The deviation of the waveform amplitude from a sinusoid presents little problems, as the signal is, typically, filtered for analogue mixing and is saturated to a square wave clock signal for digital frequency synthesisers and clocking. Thus, any amplitude variations are negligible. As it is only the phase and frequency variations that are of consequence, phase noise in oscillators is typically analysed in terms of fractional frequency deviation. The oscillator phase process can be divided into long term and short term effects. Long term effects are deterministic phase and frequency trends, such as a frequency bias and linear frequency drift, etc. Short term effects are stochastic phase and frequency processes. The oscillator phase noise process can be described as: Ψptq 2πf0 t
N ¸ 2πδfi1
i 2
i!
ti
ψ ptq,
(6.1)
where f0 and δfi represent the constant mean frequency of the oscillator and the ith
order frequency drifts, respectively [73]. The short term phase, ψ ptq, can be regarded
as a zero mean stochastic process. Any non-zero mean effects of the phase noise can be absorbed into systematic drifts. The long term quantities in (6.1) generally pose no threat to carrier tracking as any frequency bias can be absorbed into the satellite
Doppler estimate and the higher order drifts are negligible, when compared to the satellite Doppler drift. The long-term effects are, thus, neglected and, apart from f0 , assumed equal to zero, leaving only the short-term stochastic effects, ψ ptq.
The short term oscillator effects are stochastic and so cannot be represented as a sum of deterministic functions. A common method of characterising these effects is through the measurement of the phase spectrum [6]. The fractional frequency
deviation, y ptq, is defined as the instantaneous rate of change of phase with time, normalised by the nominal oscillator frequency (in radians per second): y ptq
1 δψ ptq 2πf0 δt
and the phase noise model is based on the spectrum, Sy pf q, of this process: 172
(6.2)
Section 6.1: The Local Oscillator Osc TCXO TCXO TCXO TCXO TCXO TCXO OCXO OCXO OCXO Cesium
h2 2.0e-26 2.8e-24 2.2e-26 2.4e-23 -
h1 2.5e-23 6.0e-23 6.0e-23 2.0e-23 2.9e-22 -
h0 2.0e-19 1.0e-21 9.4e-20 3.9e-22 3.5e-20 1.9e-21 2.6e-22 8.0e-20 3.4e-22 2.0e-20
h1 7.0e-21 1.0e-20 1.8e-19 2.4e-21 4.5e-20 3.5e-21 3.6e-25 2.0e-21 1.2e-21 7.0e-23
h2 2.0e-20 2.0e-20 3.8e-21 2.4e-22 8.5e-22 2.5e-23 4.0e-26 4.0e-23 1.3e-24 4.0e-29
Ref. [22] [124] [116] [80]• [118]* [118]* [80]• [22] [66]* [124]
Table 6.1: Typical Oscillator h-parameters
Sy pf q
$ 2 ¸ ' ' & α
2
hα f α
' ' %0
for fl
¤ f fh
.
(6.3)
otherwise
The coefficients hα represent the intensity of each of five noise processes and is defined in the context of a one-sided, per Hertz, power spectral density (PSD). The limits fl and fh are a function of the measurement technique and the bandwidth of the receiver [6, 73]. The coefficients represent the following noise processes: h2
white phase modulation (WPM)
h1
flicker phase modulation (FPM)
h0
white frequency modulation (WFM)(1)
h1
flicker frequency modulation (FFM)
h2
random walk frequency modulation (RWFM).
The two-sided spectrum of ψ ptq, denoted Sψ pf q, can be related to Sy pf q by [6]: Sψ pf q p2πfc q2
Sy p f q , 2f 2
(6.4)
where fc is the frequency of the tracked phase, in this case fRF . Although phase noise is generally defined in terms of the spectrum of the fractional frequency deviation, the measurement is typically a time domain measurement. The two most popular methods of measurement are Allan Variance [6] and Hadamard Variance [8]. These approaches have been generalised by [73], wherein a description of the relationship between the frequency and time domains is given. Due to the cost, size and power consumption of good quality oscillators, such as atomic frequency references, OCXOs or double oven OCXOs, consumer grade receivers typically use TCXOs. Table 6.1 shows a collection of typical oscillator hparameters for a range of typical TCXOs and, for the purposes of comparison, some OCXOs and an atomic frequency standard. The rows of Table 6.1 marked with ‘•’ 173
Chapter 6: The Phase Lock Loop II
Three−State Model Two−State Model
−8
10
Allan Deviation
σy(τ)
1 3 −9
10
2 4 −10
10
−3
10
−2
10
−1
10
0
10 τ (seconds)
1
10
2
10
3
10
Figure 6.1: Allan Deviation plots of the first four oscillators in Table 6.1. The solid lines represent three-state (h2 , h1 , h0 ) models whilst dashed lines represent two-state (h2 , h0 ) models, in which h1 is assumed to be zero.
denote referenced articles in which the oscillators were characterised in terms of (6.4),
in which the factor p2πfc q2 is absorbed into the oscillator coefficients. These values have been scaled appropriately for inclusion in Table 6.1. Also, the rows marked
with ‘*’ denote referenced articles in which h-parameters were not directly quoted and have been extracted from Allan variance charts. As WPM and FPM have very little impact on the design of carrier tracking loops in GNSS receivers (see Section 6.3), the associated h-parameters are often neglected and, consequently, not quoted. A plot of the Allan Deviation of the three frequency modulation components, WFM, FFM and RWFM, of the first four oscillators in Table 6.1, is shown in Figure 6.1. All four oscillators exhibit a similar Allan Deviation trend, having a minimum near τ
1 s averaging time and dominated by the WFM and RWFM components.
The receiver’s oscillator is not the only source of oscillator-induced phase noise; the oscillator utilized in the transmitter (the satellite) also contributes. The net phase noise is simply the linear sum of these two phase noise processes, that of the transmitter and that of the receiver. Of course, the two processes are mutually independent. Noting this independence, the Allan variance of the net phase noise, for any given averaging time, is simply the sum of the Allan variance for each phase noise source. It follows that a lumped model of the form (6.3) can be found via: hα
hTx α
Tx hRx α , where hα represents the intensity coefficient characterising the
transmitter’s oscillator and hRx α represents that of the receiver’s oscillator. In the context of GNSS systems, the hTx α coefficients are orders of magnitude smaller than the hRx α coefficients and so hα
hRx α , and, therefore, the transmitter oscillator can be 174
Section 6.1: The Local Oscillator neglected. Furthermore, in cases where a high quality receiver oscillator is employed, rendering this approximation invalid, the oscillator can no longer be assumed to be a dominant source of error, and one cannot optimise the PLL conditioned on the oscillator and thermal noise alone. For example, user dynamics may become a dominant source of error. Satellite-to-user dynamics and oscillator induced phase noise are both manifested in the same manner, as time-varying phase and frequency perturbations within the tracking loop. Both must be accurately estimated and tracked in order to maintain carrier lock. Moreover, the tracking loop cannot distinguish between the two sources of phase variation. It is convenient, therefore, to express both phenomena in the same terms, either as apparent frequency and frequency drift (Doppler and Doppler drift) or as apparent LOS dynamics. Figure 6.2 illustrates an example of apparent carrier Doppler induced by an oscillator with h-parameters equal to those of the first row of Table 6.1. Scaling this plot by the reciprocal of the carrier wavelength (which is approximately 19 cm), the apparent satellite-to-user velocity can be estimated. It can be seen in Figure 6.2 that the frequency deviation exhibits, as expected, a slowly varying component (RWFM) and a white component (WFM). Although the WFM noise can induce extremely high instantaneous apparent LOS accelerations, they are short lived (enduring for, perhaps, a number of sample periods, e.g. 1 µs) and need not all be tracked exactly by a PLL, as they may only induce small net phase deviations, which average out. Of more significance is an average acceleration sustained over a substantial time interval (comparable to the loop update period, e.g. 1 ms) as it can induce large phase and frequency errors. These variations, if they breach the linear region of the phase discriminator, can result in serious performance degradation. The dashed lines in Figure 6.2 represent regions of the apparent speed curve for which significant apparent accelerations are sustained. Here, accelerations ranging from -8 ms2 to -19 ms2 are sustained for periods of approximately 0.1 s. If not tracked, these apparent dynamics would result in phase errors of the order of
π{2 to π which would, almost certainly, result in
cycle slips and, perhaps, loss of lock.
In the context of a consumer-grade commercial application, the dynamics of a user must be considered. The characteristics of pedestrian dynamics have been well studied, both from the standpoint of bio-mechanics and of pedestrian navigation, and comprehensive models of walking and running dynamics are available [122, 69, 23, 106, 65]. A typical pedestrian will maintain an average walking speed of 1.35 ms1 [65]. Sudden turns or stops can be expected to induce accelerations of the
order of 1.45 ms2 and, in extreme cases, as large as 4.15 ms2 [23]. The dynamics of a walking stride can be expected to induce acceleration peaks and troughs of approximately 6.0 ms2 and
4.0 ms2, respectively [69].
While these dynamics
alone are negligible compared to the oscillator induced apparent dynamics, they can induce further apparent dynamics in the oscillator. 175
4
0.8
0
0 LOS Speed (ms−1)
Doppler (Hz)
Chapter 6: The Phase Lock Loop II
−4
−0.8
−8
−1.6
−12
−2.4
−16
−3.2
−20 0
5
10
15
20 Time (s)
25
30
−4 40
35
(a) Example of oscillator-induced apparent Doppler and LOS velocity.
2
0.4
−1
Doppler (Hz)
0
0
−2
LOS Speed (ms−1)
−9 ms
−0.4 −19 ms−1
−4
−0.8 −9 ms−1
−16 ms−1
−6
−1.2 2
3
4
5
6
7
Time (s) (b) A portion of the trace in (a) from 2 s to 7 s. Figure 6.2: Example of oscillator-induced apparent Doppler and oscillator-induced apparent satellite-to-user velocity for the TCXO characterised by the set of h-parameters in the first row of Table 6.1. The left and right y-axes show the apparent Doppler and satellite-to-user velocity, respectively. The dashed lines in (b) represent portions of the apparent velocity curve over which the average apparent acceleration has been estimated.
176
Section 6.2: Design of Optimal PLL Filters Due to the physical structure of crystals they are, generally, sensitive to acceleration. A quartz crystal oscillator will exhibit a slightly different series resonant frequency when subject to acceleration, when compared to the zero acceleration case. A reference signal formed using a crystal of nominal frequency f0 , subject to acceleration ~a, will exhibit a frequency:
f p~aq f0 M 1
~Γ ~a ,
(6.5)
where M is the frequency synthesiser scale factor and ~Γ is the crystal’s g-sensitivity [39]. For a typical TCXO, |~Γ| is generally of the order of 0.1 to 2.0 ppb Hz/g,
where g is the acceleration due to gravity ( 9.8 ms2 ). Taking an illustrative
example: for the GPS L1 C/A case, using a 16.3676 M Hz TCXO, the scale factor, M , is approximately 96. Assuming a g-sensitivity of 2.0 ppb, then typical pedestrian dynamics can be expected to induce rapid frequency deviations (in both positive and negative directions) of approximately 2 Hz to 3 Hz. These sudden variations are comparable to oscillator dynamics. This can be an issue for highly g-sensitive oscillators, and/or applications where the user undergoes high accelerations (running etc.) and must be explicitly addressed. In this work, however, it is neglected, and, rather, design is based on thermal and oscillator phase noise. Even in the presence of user dynamics, a loop filter conditioned on thermal noise and oscillator phase noise presents a useful lower bound on PLL bandwidth, below which the PLL will certainly under-perform. Based on these assumptions, Section 6.2 presents an optimisation of the PLL conditioned on local oscillator phase and thermal noise. Specifically, the simplified model used, similar to that of Chapter 5, is defined by:
? dm P cos pδθq ? Qm dm P sin pδθq δθ ψ rms θˆ rms em KD δθ nθ , Im
ni rms
nq rms
(6.6) (6.7) (6.8) (6.9)
and is depicted in Figure 6.3.
6.2
Design of Optimal PLL Filters
This section describes the design of a causal Wiener filter conditioned on a noisy oscillator in the presence of thermal noise. Firstly, a general approach to Wiener filter design is presented and, subsequently, this design is conditioned on the noise and oscillator parameters defined in Sections 5.1.2 and 6.1. 177
Chapter 6: The Phase Lock Loop II
nθ
ψ
δθ
+
+
KD
+
e F(z)
-
θˆ NCO(z) Figure 6.3: Linearised PLL Model used for loop optimisation.
θ(m) + θ n (m)
θˆ (m)
G(z)
Figure 6.4: Wiener Filter Block Diagram: The signal estimate, θˆpmq, is made by passing the signal-plus-noise, θpmq + θn pmq, through the filter Gpz q.
6.2.1
Wiener Filter Design
The problem definition is as follows: a discrete time signal, θpmq, is distorted by
additive white Gaussian noise, θn pmq. The distorted signal, (θpmq + θn pmq), is conditioned by a filter, Gpz q, to produce an estimate of θpmq, denoted θˆpmq. The
optimal realisation of Gpz q must minimise the estimation error, δθpmq
θpmq, in the mean square sense. This scenario is depicted in Figure 6.4.
θˆpmq
The mean square error, E, is defined as:
E
δθ2pmq D E
θpmq2 2 θpmqθˆpmq
D
E θˆpmq2 ,
(6.10)
where hxi denotes the expectation of x. Denoting the impulse response of the filter
Gpz q by g pmq, the signal estimate is given by: θˆpmq
¸
g piq pθpn iq
θn pn iqq.
(6.11)
i
The first term of (6.10) is simply the zeroth sample of the autocorrelation of θpmq,
denoted Rθ p0q. For the second term, it is noted that θpmq and θn pmq are independent, and so:
178
Section 6.2: Design of Optimal PLL Filters
D E θpmqθˆpmq
*
θ pm q
¸
¸
g piq pθpn iq
θn pn iqq
i
g piqRθ piq.
+ (6.12)
i
The third term, after some manipulation, simplifies to: D where Rθ
θn
θˆ2 pmq
E
¸¸ i
g pi qg pj qR θ
θn
pi j q ,
(6.13)
j
piq is the autocorrelation function of the distorted signal, pθpmq
θn pmqq.
Equation (6.10) can now be written in terms the filter impulse response, g pmq: E
Rθ p0q 2
¸
¸¸
g piqRθ piq
i
i
g pi qg pj qR θ
θn
pi j q .
(6.14)
j
This function must now be minimised with respect to g pmq. Using a standard
calculus of variations procedure, g pmq is replaced by a perturbed weighting function g pmq
εη pmq [22]. By definition, E will have a minimum at ε 0. Thus, differen-
tiating with respect to ε and equating the result to zero at ε 0, yields, after some
manipulation:
¸
η piq
¸
Rθ piq
pi j q 0.
(6.15)
In finding a causal solution for g pmq, it is assumed that g pmq
εη pmq is zero for
i
8 m 0. then:
g piqRθ
θn
j
Thus, if apmq is an arbitrary function which is zero for 0 ¸
Rθ piq
g piqRθ
θn
pi j q a pi q.
¤ m ¤ 8,
(6.16)
j
Taking the z-transforms of (6.16) yields: G pz qS θ
pzq Sθ pzq Apzq, (6.17) where Sθ θ pz q and Sθ pz q are the z-transforms of Rθ θ pmq and Rθ pmq, respectively. If Sθ θ pz q and Sθ pz q are now factored into positive-time and negative-time spectral θn
n
n
n
densities, where superscripts ‘+’ and ‘-’ denote positive- and negative-time parts, respectively, then:
G pz qS θ
θn
pzq S Apzpqzq θ θn
Sθ pz q . θ θn pz q
S
(6.18)
The left hand side of (6.18) is a positive-time only function, the first term on the right hand side is a negative-time only function and the second term on the right hand side is both a positive and negative time function. Equating positive- and 179
Chapter 6: The Phase Lock Loop II negative-time parts and solving for Gpz q yields a solution for the optimum filter, in
terms of known spectral densities: Gp z q
Sθ pz q PT , Sθ θn pz q Sθ θn pz q 1
(6.19)
where the operator PTrX pz qs denotes the positive-time part of X pz q.
It is now necessary to infer the optimal PLL loop filter, F pz q, from this result.
The premise of the Wiener filter design was that an unbiased estimate of the signal carrier phase, θpmq, corrupted by a white noise, θn pmq, is available to the filter. In the case of the GNSS PLL, a biased estimate is available, in the form of the dis-
criminator output, e. To conform with (6.19), the estimate must first be normalised by the discriminator gain, KD . The result of this normalisation is the following equivalence of the PLL signals and those of the Wiener filter:
θ pm q θ n pm q
e KD nθ pmq . KD
(6.20) (6.21)
Furthermore, as it is intended to design a digital filter which is conditioned on
a known spectral density, it is convenient that SΨ pf q be expressed as an equivalent
discrete-time spectral density. Also, as it is intended to factor this spectrum, as in (6.19), it is necessary that this discrete-time spectrum take the form H pz qH pz 1 q,
where H pz q is a rational function of integer powers of z. To satisfy this requirement,
the equivalence of backward-difference integration approximations and analogue integrators (i.e. the equivalence of the Laplace Transform of an integration,
and the z-transform of an accumulation,
z z 1F
p2πqn p2πqn f 2n sn psqn
n 2πz n 2πz 1 z1 z 1 1 1
1 sF
ps q,
pzq), can be utilised [103]. That is: s Ñ i2πf
(6.22)
Ñ ei2πf
(6.23)
z
such that, from (6.4):
p2πfcq2 Sˆψ pz q 2
h0z h2 z 2 pz 1q2 pz 1q4
(6.24)
where Sˆψ pz q denotes the discrete-time approximation to Sψ pf q. Note that the spec-
tral components with coefficients h2 , h1 and h1 have been omitted. The reasons
for this will be discussed next. The PSD of the signal and of the noise incident on 180
Section 6.2: Design of Optimal PLL Filters the PLL filter can, thus, be related to those of the Wiener filter via:
Sθ pz q Sˆψ pz q Sθn pz q Sθ
θn
θ
Var n TL 2 KD
p z q S θ pz q
Sθ
(6.25) TL Nθ GNR
n
θn
pz q .
(6.26) (6.27)
Interestingly, the metric GNR, introduced in Chapter 5, reappears in (6.26) in the context of the phase estimate noise floor. For expository simplicity, the variable Nθn is defined to denote this noise floor.
ˆ pz q As the transfer function Gpz q represents the optimal relationship between Θ
and Θpz q, it can be related to (5.8) to find the PLL filter. As indicated in (6.20), the gain degradation of the discriminator estimate must be countered. Doing so
by scaling the discriminator estimate is equivalent to scaling the filter, F pz q, itself. This effectively removes the dependence of the transfer function Hθˆpz q on KD and so the following relationship can be utilized:
G pz q F pz q
N COpz qF pz q 1 N COpz qF pz q G pz q . N COpz q p1 Gpz qq
(6.28) (6.29)
Before proceeding with a practical implementation of a Wiener filter, it is worthwhile to consider the general principle of the filter and some of its obvious limitations. The Wiener filter, appropriately designed, is a filter which provides optimal separation of a signal-plus-noise process such that the filtered signal is the MMSE estimate of the original signal. When the signal and noise spectra overlap, which is normally the case, this filter will effect the optimal trade-off between maximisation of noise rejection and minimisation of signal attenuation and distortion. One prerequisite of this, of course, is that there is some spectral separation between signal and noise, that is, the signal PSD cannot be expressed as a scalar times the noise spectrum. If the PSD of the signal and noise are identical, then any attempt to design a filter based on these spectra will result in either undefined, or trivial, solutions. For example, from (6.4), one component of the oscillator noise model is a white phase modulation, which has a white PSD with a (two-sided) spectral density of h2 . Given that thermal noise also has a white PSD, it is not possible to achieve any separation of signal and noise using an LTI filter. Some discretion must be used, therefore, when choosing the PSD, Sθ pz q, when performing a Wiener filter design.
Another factor which must be considered is that of the properties of the signal PSD. As WPM is, by definition, a zero mean white process, it cannot be tracked. 181
Chapter 6: The Phase Lock Loop II While a filter may prove useful in separating a sample of a white signal from a non-white noise, for the purposes of phase tracking, no advantage can be gained from considering WPM in the PLL design. WPM cannot be neglected entirely, however, and a designer must ensure that h2 is sufficiently low so as not to contribute to a degradation in the quality of the correlator values, Im and Qm [97]. From examination of Table 6.1, it is clear that this limitation on h2 , however, is readily satisfied by typical oscillators. Finally, it must be ensured that the filter is realisable. Similar to the constraint
imposed on g pmq
εη pmq (that it is zero for negative m) in the derivation of the
Wiener filter, some constraints must also be placed on the signal and noise spectra.
As Gpz q is evaluated from the ratios Sθ pz q and Sθn pz q, if either are not simple and
real functions of z, then they may lead to non-realisable filter designs. In particular, FPM and FFM pose problems. Being, so called, fractional frequency processes, their PSDs cannot be expressed in terms of integer powers of f or z. The result of Wiener filters conditioned on fractional spectra, can have fractional spectra, rendering them, for all practical purposes, useless. As discussed here, and in Section 6.1, a reduced phase noise model must, therefore, be used for filter design. The next section will begin by designing a Wiener filter conditioned on WFM only and, subsequently, on both WFM and RWFM. The assumption, that the oscillator dominates the received phase anomaly, is explored and it is found to be valid only under certain conditions. It will be illustrated that when the oscillator model is simplified beyond a certain point, the resulting filter is first order and, therefore, incapable of sustaining Doppler-induced phase effects. The limitations of the Wiener design are examined and a modification to the pure Wiener filter is proposed.
6.2.2
White Frequency Noise
Using (6.25) and (6.26), and neglecting all phase noise coefficients other than h0 , the signal and signal plus noise PSDs are given by:
Sθ p z q
p2πfcq2 h20z k02z 2 pz 1 q2 pz 1 q2
Sθn pz q Nθn Sθ
θn
pzq pz 1pzq 1q2 2N θn
(6.30) (6.31)
k02 z
(6.32)
.
Not that the notation has been simplified somewhat here by introducing the variable k0 , which accommodates the scale factor 182
p2πfc q2 . S θ 2
θn
pz q
can be factorised by
Section 6.2: Design of Optimal PLL Filters assuming a function B pz q such that: Sθ
pz q B pz q B pz 1 q a pz z b q B pz q p z 1q ,
θn
(6.33)
where a and zb are constants. Solving directly for ta, zb u yields the following set of four solutions:
1 pk0 κq ,zb 2 1 pk0 κq ,zb 2
1
1 pk0 κq ,zb 2 1 pk0 κq ,zb 2
1
a a a a
k0 2Nθn k0 2Nθn k0 2Nθn k0 2Nθn
1 1
pk0 κq
(
pk0
κq
(
pk0 κq
(
pk0
(
κq ,
(6.34) (6.35) (6.36) (6.37)
where the notation has been simplified by the introduction of the intermediate variable, κ
a
4Nθn
k02 . This set of four solutions can be reduced to one unique
solution by stipulating that a ¡ 0, a P
